/*
 * Copyright (c) 2023, Alibaba Group Holding Limited;
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
// Copyright 2020-2022 Junekey Jeon
//
// The contents of this file may be used under the terms of
// the Apache License v2.0 with LLVM Exceptions.
//
//    (See accompanying file LICENSE-Apache or copy at
//     https://llvm.org/foundation/relicensing/LICENSE.txt)
//
// Alternatively, the contents of this file may be used under the terms of
// the Boost Software License, Version 1.0.
//    (See accompanying file LICENSE-Boost or copy at
//     https://www.boost.org/LICENSE_1_0.txt)
//
// Unless required by applicable law or agreed to in writing, this software
// is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
// KIND, either express or implied.

// commit-id: 33a9e021290d529bcb41773be2c7c3c91726a9cb

#ifndef JKJ_HEADER_DRAGONBOX
#define JKJ_HEADER_DRAGONBOX

#include <cassert>
#include <cstdint>
#include <cstring>
#include <limits>
#include <type_traits>

// Suppress additional buffer overrun check.
// I have no idea why MSVC thinks some functions here are vulnerable to the
// buffer overrun attacks. No, they aren't.
#if defined(__GNUC__) || defined(__clang__)
#define JKJ_SAFEBUFFERS
#define JKJ_FORCEINLINE inline __attribute__((always_inline))
#elif defined(_MSC_VER)
#define JKJ_SAFEBUFFERS __declspec(safebuffers)
#define JKJ_FORCEINLINE __forceinline
#else
#define JKJ_SAFEBUFFERS
#define JKJ_FORCEINLINE inline
#endif

#if defined(__has_builtin)
#define JKJ_DRAGONBOX_HAS_BUILTIN(x) __has_builtin(x)
#else
#define JKJ_DRAGONBOX_HAS_BUILTIN(x) false
#endif

#if defined(_MSC_VER)
#include <intrin.h>
#endif

namespace jkj::dragonbox {
namespace detail {
template <class T>
constexpr std::size_t physical_bits =
    sizeof(T) * std::numeric_limits<unsigned char>::digits;

template <class T>
constexpr std::size_t value_bits =
    std::numeric_limits<std::enable_if_t<std::is_unsigned_v<T>, T>>::digits;
}  // namespace detail

// These classes expose encoding specs of IEEE-754-like floating-point formats.
// Currently available formats are IEEE754-binary32 & IEEE754-binary64.

struct ieee754_binary32 {
  static constexpr int significand_bits = 23;
  static constexpr int exponent_bits = 8;
  static constexpr int min_exponent = -126;
  static constexpr int max_exponent = 127;
  static constexpr int exponent_bias = -127;
  static constexpr int decimal_digits = 9;
};
struct ieee754_binary64 {
  static constexpr int significand_bits = 52;
  static constexpr int exponent_bits = 11;
  static constexpr int min_exponent = -1022;
  static constexpr int max_exponent = 1023;
  static constexpr int exponent_bias = -1023;
  static constexpr int decimal_digits = 17;
};

// A floating-point traits class defines ways to interpret a bit pattern of
// given size as an encoding of floating-point number. This is a default
// implementation of such a traits class, supporting ways to interpret 32-bits
// into a binary32-encoded floating-point number and to interpret 64-bits into a
// binary64-encoded floating-point number. Users might specialize this class to
// change the default behavior for certain types.
template <class T>
struct default_float_traits {
  // I don't know if there is a truly reliable way of detecting
  // IEEE-754 binary32/binary64 formats; I just did my best here.
  static_assert(
      std::numeric_limits<T>::is_iec559 && std::numeric_limits<T>::radix == 2 &&
          (detail::physical_bits<T> == 32 || detail::physical_bits<T> == 64),
      "default_ieee754_traits only works for 32-bits or 64-bits types "
      "supporting binary32 or binary64 formats!");

  // The type that is being viewed.
  using type = T;

  // Refers to the format specification class.
  using format = std::conditional_t<detail::physical_bits<T> == 32,
                                    ieee754_binary32, ieee754_binary64>;

  // Defines an unsigned integer type that is large enough to carry a variable
  // of type T. Most of the operations will be done on this integer type.
  using carrier_uint = std::conditional_t<detail::physical_bits<T> == 32,
                                          std::uint32_t, std::uint64_t>;
  static_assert(sizeof(carrier_uint) == sizeof(T));

  // Number of bits in the above unsigned integer type.
  static constexpr int carrier_bits = int(detail::physical_bits<carrier_uint>);

  // Convert from carrier_uint into the original type.
  // Depending on the floating-point encoding format, this operation might not
  // be possible for some specific bit patterns. However, the contract is that u
  // always denotes a valid bit pattern, so this function must be assumed to be
  // noexcept.
  static T carrier_to_float(carrier_uint u) noexcept {
    T x;
    std::memcpy(&x, &u, sizeof(carrier_uint));
    return x;
  }

  // Same as above.
  static carrier_uint float_to_carrier(T x) noexcept {
    carrier_uint u;
    std::memcpy(&u, &x, sizeof(carrier_uint));
    return u;
  }

  // Extract exponent bits from a bit pattern.
  // The result must be aligned to the LSB so that there is no additional zero
  // paddings on the right. This function does not do bias adjustment.
  static constexpr unsigned int extract_exponent_bits(carrier_uint u) noexcept {
    constexpr int significand_bits = format::significand_bits;
    constexpr int exponent_bits = format::exponent_bits;
    static_assert(detail::value_bits<unsigned int> > exponent_bits);
    constexpr auto exponent_bits_mask =
        (unsigned int)(((unsigned int)(1) << exponent_bits) - 1);
    return (unsigned int)(u >> significand_bits) & exponent_bits_mask;
  }

  // Extract significand bits from a bit pattern.
  // The result must be aligned to the LSB so that there is no additional zero
  // paddings on the right. The result does not contain the implicit bit.
  static constexpr carrier_uint extract_significand_bits(
      carrier_uint u) noexcept {
    constexpr auto mask =
        carrier_uint((carrier_uint(1) << format::significand_bits) - 1);
    return carrier_uint(u & mask);
  }

  // Remove the exponent bits and extract significand bits together with the
  // sign bit.
  static constexpr carrier_uint remove_exponent_bits(
      carrier_uint u, unsigned int exponent_bits) noexcept {
    return u ^ (carrier_uint(exponent_bits) << format::significand_bits);
  }

  // Shift the obtained signed significand bits to the left by 1 to remove the
  // sign bit.
  static constexpr carrier_uint remove_sign_bit_and_shift(
      carrier_uint u) noexcept {
    return carrier_uint(carrier_uint(u) << 1);
  }

  // The actual value of exponent is obtained by adding this value to the
  // extracted exponent bits.
  static constexpr int exponent_bias =
      1 - (1 << (carrier_bits - format::significand_bits - 2));

  // Obtain the actual value of the binary exponent from the extracted exponent
  // bits.
  static constexpr int binary_exponent(unsigned int exponent_bits) noexcept {
    if (exponent_bits == 0) {
      return format::min_exponent;
    }
    else {
      return int(exponent_bits) + format::exponent_bias;
    }
  }

  // Obtain the actual value of the binary exponent from the extracted
  // significand bits and exponent bits.
  static constexpr carrier_uint binary_significand(
      carrier_uint significand_bits, unsigned int exponent_bits) noexcept {
    if (exponent_bits == 0) {
      return significand_bits;
    }
    else {
      return significand_bits | (carrier_uint(1) << format::significand_bits);
    }
  }

  /* Various boolean observer functions */

  static constexpr bool is_nonzero(carrier_uint u) noexcept {
    return (u << 1) != 0;
  }
  static constexpr bool is_positive(carrier_uint u) noexcept {
    constexpr auto sign_bit =
        carrier_uint(1) << (format::significand_bits + format::exponent_bits);
    return u < sign_bit;
  }
  static constexpr bool is_negative(carrier_uint u) noexcept {
    return !is_positive(u);
  }
  static constexpr bool is_finite(unsigned int exponent_bits) noexcept {
    constexpr unsigned int exponent_bits_all_set =
        (1u << format::exponent_bits) - 1;
    return exponent_bits != exponent_bits_all_set;
  }
  static constexpr bool has_all_zero_significand_bits(carrier_uint u) noexcept {
    return (u << 1) == 0;
  }
  static constexpr bool has_even_significand_bits(carrier_uint u) noexcept {
    return u % 2 == 0;
  }
};

// Convenient wrappers for floating-point traits classes.
// In order to reduce the argument passing overhead, these classes should be as
// simple as possible (e.g., no inheritance, no private non-static data member,
// etc.; this is an unfortunate fact about common ABI convention).

template <class T, class Traits = default_float_traits<T>>
struct float_bits;

template <class T, class Traits = default_float_traits<T>>
struct signed_significand_bits;

template <class T, class Traits>
struct float_bits {
  using type = T;
  using traits_type = Traits;
  using carrier_uint = typename traits_type::carrier_uint;

  carrier_uint u;

  float_bits() = default;
  constexpr explicit float_bits(carrier_uint bit_pattern) noexcept
      : u{bit_pattern} {}
  constexpr explicit float_bits(T float_value) noexcept
      : u{traits_type::float_to_carrier(float_value)} {}

  constexpr T to_float() const noexcept {
    return traits_type::carrier_to_float(u);
  }

  // Extract exponent bits from a bit pattern.
  // The result must be aligned to the LSB so that there is no additional zero
  // paddings on the right. This function does not do bias adjustment.
  constexpr unsigned int extract_exponent_bits() const noexcept {
    return traits_type::extract_exponent_bits(u);
  }

  // Extract significand bits from a bit pattern.
  // The result must be aligned to the LSB so that there is no additional zero
  // paddings on the right. The result does not contain the implicit bit.
  constexpr carrier_uint extract_significand_bits() const noexcept {
    return traits_type::extract_significand_bits(u);
  }

  // Remove the exponent bits and extract significand bits together with the
  // sign bit.
  constexpr auto remove_exponent_bits(
      unsigned int exponent_bits) const noexcept {
    return signed_significand_bits<type, traits_type>(
        traits_type::remove_exponent_bits(u, exponent_bits));
  }

  // Obtain the actual value of the binary exponent from the extracted exponent
  // bits.
  static constexpr int binary_exponent(unsigned int exponent_bits) noexcept {
    return traits_type::binary_exponent(exponent_bits);
  }
  constexpr int binary_exponent() const noexcept {
    return binary_exponent(extract_exponent_bits());
  }

  // Obtain the actual value of the binary exponent from the extracted
  // significand bits and exponent bits.
  static constexpr carrier_uint binary_significand(
      carrier_uint significand_bits, unsigned int exponent_bits) noexcept {
    return traits_type::binary_significand(significand_bits, exponent_bits);
  }
  constexpr carrier_uint binary_significand() const noexcept {
    return binary_significand(extract_significand_bits(),
                              extract_exponent_bits());
  }

  constexpr bool is_nonzero() const noexcept {
    return traits_type::is_nonzero(u);
  }
  constexpr bool is_positive() const noexcept {
    return traits_type::is_positive(u);
  }
  constexpr bool is_negative() const noexcept {
    return traits_type::is_negative(u);
  }
  constexpr bool is_finite(unsigned int exponent_bits) const noexcept {
    return traits_type::is_finite(exponent_bits);
  }
  constexpr bool is_finite() const noexcept {
    return traits_type::is_finite(extract_exponent_bits());
  }
  constexpr bool has_even_significand_bits() const noexcept {
    return traits_type::has_even_significand_bits(u);
  }
};

template <class T, class Traits>
struct signed_significand_bits {
  using type = T;
  using traits_type = Traits;
  using carrier_uint = typename traits_type::carrier_uint;

  carrier_uint u;

  signed_significand_bits() = default;
  constexpr explicit signed_significand_bits(carrier_uint bit_pattern) noexcept
      : u{bit_pattern} {}

  // Shift the obtained signed significand bits to the left by 1 to remove the
  // sign bit.
  constexpr carrier_uint remove_sign_bit_and_shift() const noexcept {
    return traits_type::remove_sign_bit_and_shift(u);
  }

  constexpr bool is_positive() const noexcept {
    return traits_type::is_positive(u);
  }
  constexpr bool is_negative() const noexcept {
    return traits_type::is_negative(u);
  }
  constexpr bool has_all_zero_significand_bits() const noexcept {
    return traits_type::has_all_zero_significand_bits(u);
  }
  constexpr bool has_even_significand_bits() const noexcept {
    return traits_type::has_even_significand_bits(u);
  }
};

namespace detail {
////////////////////////////////////////////////////////////////////////////////////////
// Bit operation intrinsics.
////////////////////////////////////////////////////////////////////////////////////////

namespace bits {
// Most compilers should be able to optimize this into the ROR instruction.
inline std::uint32_t rotr(std::uint32_t n, std::uint32_t r) noexcept {
  r &= 31;
  return (n >> r) | (n << (32 - r));
}
inline std::uint64_t rotr(std::uint64_t n, std::uint32_t r) noexcept {
  r &= 63;
  return (n >> r) | (n << (64 - r));
}
}  // namespace bits

////////////////////////////////////////////////////////////////////////////////////////
// Utilities for wide unsigned integer arithmetic.
////////////////////////////////////////////////////////////////////////////////////////

namespace wuint {
// Compilers might support built-in 128-bit integer types. However, it seems
// that emulating them with a pair of 64-bit integers actually produces a better
// code, so we avoid using those built-ins. That said, they are still useful for
// implementing 64-bit x 64-bit -> 128-bit multiplication.

// clang-format off
#if defined(__SIZEOF_INT128__)
		// To silence "error: ISO C++ does not support '__int128' for 'type name'
		// [-Wpedantic]"
#if defined(__GNUC__)
			__extension__
#endif
				using builtin_uint128_t = unsigned __int128;
#endif
// clang-format on

struct uint128 {
  uint128() = default;

  std::uint64_t high_;
  std::uint64_t low_;

  constexpr uint128(std::uint64_t high, std::uint64_t low) noexcept
      : high_{high}, low_{low} {}

  constexpr std::uint64_t high() const noexcept { return high_; }
  constexpr std::uint64_t low() const noexcept { return low_; }

  uint128 &operator+=(std::uint64_t n) &noexcept {
#if JKJ_DRAGONBOX_HAS_BUILTIN(__builtin_addcll)
    unsigned long long carry;
    low_ = __builtin_addcll(low_, n, 0, &carry);
    high_ = __builtin_addcll(high_, 0, carry, &carry);
#elif JKJ_DRAGONBOX_HAS_BUILTIN(__builtin_ia32_addcarryx_u64)
    unsigned long long result;
    auto carry = __builtin_ia32_addcarryx_u64(0, low_, n, &result);
    low_ = result;
    __builtin_ia32_addcarryx_u64(carry, high_, 0, &result);
    high_ = result;
#elif defined(_MSC_VER) && defined(_M_X64)
    auto carry = _addcarry_u64(0, low_, n, &low_);
    _addcarry_u64(carry, high_, 0, &high_);
#else
    auto sum = low_ + n;
    high_ += (sum < low_ ? 1 : 0);
    low_ = sum;
#endif
    return *this;
  }
};

static inline std::uint64_t umul64(std::uint32_t x, std::uint32_t y) noexcept {
#if defined(_MSC_VER) && defined(_M_IX86)
  return __emulu(x, y);
#else
  return x * std::uint64_t(y);
#endif
}

// Get 128-bit result of multiplication of two 64-bit unsigned integers.
JKJ_SAFEBUFFERS inline uint128 umul128(std::uint64_t x,
                                       std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__)
  auto result = builtin_uint128_t(x) * builtin_uint128_t(y);
  return {std::uint64_t(result >> 64), std::uint64_t(result)};
#elif defined(_MSC_VER) && defined(_M_X64)
  uint128 result;
  result.low_ = _umul128(x, y, &result.high_);
  return result;
#else
  auto a = std::uint32_t(x >> 32);
  auto b = std::uint32_t(x);
  auto c = std::uint32_t(y >> 32);
  auto d = std::uint32_t(y);

  auto ac = umul64(a, c);
  auto bc = umul64(b, c);
  auto ad = umul64(a, d);
  auto bd = umul64(b, d);

  auto intermediate = (bd >> 32) + std::uint32_t(ad) + std::uint32_t(bc);

  return {ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32),
          (intermediate << 32) + std::uint32_t(bd)};
#endif
}

JKJ_SAFEBUFFERS inline std::uint64_t umul128_upper64(std::uint64_t x,
                                                     std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__)
  auto result = builtin_uint128_t(x) * builtin_uint128_t(y);
  return std::uint64_t(result >> 64);
#elif defined(_MSC_VER) && defined(_M_X64)
  return __umulh(x, y);
#else
  auto a = std::uint32_t(x >> 32);
  auto b = std::uint32_t(x);
  auto c = std::uint32_t(y >> 32);
  auto d = std::uint32_t(y);

  auto ac = umul64(a, c);
  auto bc = umul64(b, c);
  auto ad = umul64(a, d);
  auto bd = umul64(b, d);

  auto intermediate = (bd >> 32) + std::uint32_t(ad) + std::uint32_t(bc);

  return ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32);
#endif
}

// Get upper 128-bits of multiplication of a 64-bit unsigned integer and a
// 128-bit unsigned integer.
JKJ_SAFEBUFFERS inline uint128 umul192_upper128(std::uint64_t x,
                                                uint128 y) noexcept {
  auto r = umul128(x, y.high());
  r += umul128_upper64(x, y.low());
  return r;
}

// Get upper 64-bits of multiplication of a 32-bit unsigned integer and a 64-bit
// unsigned integer.
inline std::uint64_t umul96_upper64(std::uint32_t x, std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__) || (defined(_MSC_VER) && defined(_M_X64))
  return umul128_upper64(std::uint64_t(x) << 32, y);
#else
  auto yh = std::uint32_t(y >> 32);
  auto yl = std::uint32_t(y);

  auto xyh = umul64(x, yh);
  auto xyl = umul64(x, yl);

  return xyh + (xyl >> 32);
#endif
}

// Get lower 128-bits of multiplication of a 64-bit unsigned integer and a
// 128-bit unsigned integer.
JKJ_SAFEBUFFERS inline uint128 umul192_lower128(std::uint64_t x,
                                                uint128 y) noexcept {
  auto high = x * y.high();
  auto high_low = umul128(x, y.low());
  return {high + high_low.high(), high_low.low()};
}

// Get lower 64-bits of multiplication of a 32-bit unsigned integer and a 64-bit
// unsigned integer.
inline std::uint64_t umul96_lower64(std::uint32_t x, std::uint64_t y) noexcept {
  return x * y;
}
}  // namespace wuint

////////////////////////////////////////////////////////////////////////////////////////
// Some simple utilities for constexpr computation.
////////////////////////////////////////////////////////////////////////////////////////

template <int k, class Int>
constexpr Int compute_power(Int a) noexcept {
  static_assert(k >= 0);
  Int p = 1;
  for (int i = 0; i < k; ++i) {
    p *= a;
  }
  return p;
}

template <int a, class UInt>
constexpr int count_factors(UInt n) noexcept {
  static_assert(a > 1);
  int c = 0;
  while (n % a == 0) {
    n /= a;
    ++c;
  }
  return c;
}

////////////////////////////////////////////////////////////////////////////////////////
// Utilities for fast/constexpr log computation.
////////////////////////////////////////////////////////////////////////////////////////

namespace log {
static_assert((-1 >> 1) == -1,
              "right-shift for signed integers must be arithmetic");

// Compute floor(e * c - s).
enum class multiply : std::uint32_t {};
enum class subtract : std::uint32_t {};
enum class shift : std::size_t {};
enum class min_exponent : std::int32_t {};
enum class max_exponent : std::int32_t {};

template <multiply m, subtract f, shift k, min_exponent e_min,
          max_exponent e_max>
constexpr int compute(int e) noexcept {
  assert(std::int32_t(e_min) <= e && e <= std::int32_t(e_max));
  return int((std::int32_t(e) * std::int32_t(m) - std::int32_t(f)) >>
             std::size_t(k));
}

// For constexpr computation.
// Returns -1 when n = 0.
template <class UInt>
constexpr int floor_log2(UInt n) noexcept {
  int count = -1;
  while (n != 0) {
    ++count;
    n >>= 1;
  }
  return count;
}

static constexpr int floor_log10_pow2_min_exponent = -2620;
static constexpr int floor_log10_pow2_max_exponent = 2620;
constexpr int floor_log10_pow2(int e) noexcept {
  using namespace log;
  return compute<multiply(315653), subtract(0), shift(20),
                 min_exponent(floor_log10_pow2_min_exponent),
                 max_exponent(floor_log10_pow2_max_exponent)>(e);
}

static constexpr int floor_log2_pow10_min_exponent = -1233;
static constexpr int floor_log2_pow10_max_exponent = 1233;
constexpr int floor_log2_pow10(int e) noexcept {
  using namespace log;
  return compute<multiply(1741647), subtract(0), shift(19),
                 min_exponent(floor_log2_pow10_min_exponent),
                 max_exponent(floor_log2_pow10_max_exponent)>(e);
}

static constexpr int floor_log10_pow2_minus_log10_4_over_3_min_exponent = -2985;
static constexpr int floor_log10_pow2_minus_log10_4_over_3_max_exponent = 2936;
constexpr int floor_log10_pow2_minus_log10_4_over_3(int e) noexcept {
  using namespace log;
  return compute<
      multiply(631305), subtract(261663), shift(21),
      min_exponent(floor_log10_pow2_minus_log10_4_over_3_min_exponent),
      max_exponent(floor_log10_pow2_minus_log10_4_over_3_max_exponent)>(e);
}

static constexpr int floor_log5_pow2_min_exponent = -1831;
static constexpr int floor_log5_pow2_max_exponent = 1831;
constexpr int floor_log5_pow2(int e) noexcept {
  using namespace log;
  return compute<multiply(225799), subtract(0), shift(19),
                 min_exponent(floor_log5_pow2_min_exponent),
                 max_exponent(floor_log5_pow2_max_exponent)>(e);
}

static constexpr int floor_log5_pow2_minus_log5_3_min_exponent = -3543;
static constexpr int floor_log5_pow2_minus_log5_3_max_exponent = 2427;
constexpr int floor_log5_pow2_minus_log5_3(int e) noexcept {
  using namespace log;
  return compute<multiply(451597), subtract(715764), shift(20),
                 min_exponent(floor_log5_pow2_minus_log5_3_min_exponent),
                 max_exponent(floor_log5_pow2_minus_log5_3_max_exponent)>(e);
}
}  // namespace log

////////////////////////////////////////////////////////////////////////////////////////
// Utilities for fast divisibility tests.
////////////////////////////////////////////////////////////////////////////////////////

namespace div {
// Replace n by floor(n / 10^N).
// Returns true if and only if n is divisible by 10^N.
// Precondition: n <= 10^(N+1)
// !!It takes an in-out parameter!!
template <int N>
struct divide_by_pow10_info;

template <>
struct divide_by_pow10_info<1> {
  static constexpr std::uint32_t magic_number = 6554;
  static constexpr int shift_amount = 16;
};

template <>
struct divide_by_pow10_info<2> {
  static constexpr std::uint32_t magic_number = 656;
  static constexpr int shift_amount = 16;
};

template <int N>
constexpr bool check_divisibility_and_divide_by_pow10(
    std::uint32_t &n) noexcept {
  // Make sure the computation for max_n does not overflow.
  static_assert(N + 1 <= log::floor_log10_pow2(31));
  assert(n <= compute_power<N + 1>(std::uint32_t(10)));

  using info = divide_by_pow10_info<N>;
  n *= info::magic_number;

  constexpr auto mask =
      std::uint32_t(std::uint32_t(1) << info::shift_amount) - 1;
  bool result = ((n & mask) < info::magic_number);

  n >>= info::shift_amount;
  return result;
}

// Compute floor(n / 10^N) for small n and N.
// Precondition: n <= 10^(N+1)
template <int N>
constexpr std::uint32_t small_division_by_pow10(std::uint32_t n) noexcept {
  // Make sure the computation for max_n does not overflow.
  static_assert(N + 1 <= log::floor_log10_pow2(31));
  assert(n <= compute_power<N + 1>(std::uint32_t(10)));

  return (n * divide_by_pow10_info<N>::magic_number) >>
         divide_by_pow10_info<N>::shift_amount;
}

// Compute floor(n / 10^N) for small N.
// Precondition: n <= n_max
template <int N, class UInt, UInt n_max>
constexpr UInt divide_by_pow10(UInt n) noexcept {
  static_assert(N >= 0);

  // Specialize for 32-bit division by 100.
  // Compiler is supposed to generate the identical code for just writing
  // "n / 100", but for some reason MSVC generates an inefficient code
  // (mul + mov for no apparent reason, instead of single imul),
  // so we does this manually.
  if constexpr (std::is_same_v<UInt, std::uint32_t> && N == 2) {
    return std::uint32_t(wuint::umul64(n, std::uint32_t(1374389535)) >> 37);
  }
  // Specialize for 64-bit division by 1000.
  // Ensure that the correctness condition is met.
  if constexpr (std::is_same_v<UInt, std::uint64_t> && N == 3 &&
                n_max <= std::uint64_t(15534100272597517998ull)) {
    return wuint::umul128_upper64(n, std::uint64_t(2361183241434822607ull)) >>
           7;
  }
  else {
    constexpr auto divisor = compute_power<N>(UInt(10));
    return n / divisor;
  }
}
}  // namespace div
}  // namespace detail

////////////////////////////////////////////////////////////////////////////////////////
// Return types for the main interface function.
////////////////////////////////////////////////////////////////////////////////////////

template <class UInt, bool is_signed, bool trailing_zero_flag>
struct decimal_fp;

template <class UInt>
struct decimal_fp<UInt, false, false> {
  using carrier_uint = UInt;

  carrier_uint significand;
  int exponent;
};

template <class UInt>
struct decimal_fp<UInt, true, false> {
  using carrier_uint = UInt;

  carrier_uint significand;
  int exponent;
  bool is_negative;
};

template <class UInt>
struct decimal_fp<UInt, false, true> {
  using carrier_uint = UInt;

  carrier_uint significand;
  int exponent;
  bool may_have_trailing_zeros;
};

template <class UInt>
struct decimal_fp<UInt, true, true> {
  using carrier_uint = UInt;

  carrier_uint significand;
  int exponent;
  bool is_negative;
  bool may_have_trailing_zeros;
};

template <class UInt>
using unsigned_decimal_fp = decimal_fp<UInt, false, false>;

template <class UInt>
using signed_decimal_fp = decimal_fp<UInt, true, false>;

////////////////////////////////////////////////////////////////////////////////////////
// Computed cache entries.
////////////////////////////////////////////////////////////////////////////////////////

namespace detail {
template <class FloatFormat>
struct cache_holder;

template <>
struct cache_holder<ieee754_binary32> {
  using cache_entry_type = std::uint64_t;
  static constexpr int cache_bits = 64;
  static constexpr int min_k = -31;
  static constexpr int max_k = 46;
  static constexpr cache_entry_type cache[] = {
      0x81ceb32c4b43fcf5, 0xa2425ff75e14fc32, 0xcad2f7f5359a3b3f,
      0xfd87b5f28300ca0e, 0x9e74d1b791e07e49, 0xc612062576589ddb,
      0xf79687aed3eec552, 0x9abe14cd44753b53, 0xc16d9a0095928a28,
      0xf1c90080baf72cb2, 0x971da05074da7bef, 0xbce5086492111aeb,
      0xec1e4a7db69561a6, 0x9392ee8e921d5d08, 0xb877aa3236a4b44a,
      0xe69594bec44de15c, 0x901d7cf73ab0acda, 0xb424dc35095cd810,
      0xe12e13424bb40e14, 0x8cbccc096f5088cc, 0xafebff0bcb24aaff,
      0xdbe6fecebdedd5bf, 0x89705f4136b4a598, 0xabcc77118461cefd,
      0xd6bf94d5e57a42bd, 0x8637bd05af6c69b6, 0xa7c5ac471b478424,
      0xd1b71758e219652c, 0x83126e978d4fdf3c, 0xa3d70a3d70a3d70b,
      0xcccccccccccccccd, 0x8000000000000000, 0xa000000000000000,
      0xc800000000000000, 0xfa00000000000000, 0x9c40000000000000,
      0xc350000000000000, 0xf424000000000000, 0x9896800000000000,
      0xbebc200000000000, 0xee6b280000000000, 0x9502f90000000000,
      0xba43b74000000000, 0xe8d4a51000000000, 0x9184e72a00000000,
      0xb5e620f480000000, 0xe35fa931a0000000, 0x8e1bc9bf04000000,
      0xb1a2bc2ec5000000, 0xde0b6b3a76400000, 0x8ac7230489e80000,
      0xad78ebc5ac620000, 0xd8d726b7177a8000, 0x878678326eac9000,
      0xa968163f0a57b400, 0xd3c21bcecceda100, 0x84595161401484a0,
      0xa56fa5b99019a5c8, 0xcecb8f27f4200f3a, 0x813f3978f8940985,
      0xa18f07d736b90be6, 0xc9f2c9cd04674edf, 0xfc6f7c4045812297,
      0x9dc5ada82b70b59e, 0xc5371912364ce306, 0xf684df56c3e01bc7,
      0x9a130b963a6c115d, 0xc097ce7bc90715b4, 0xf0bdc21abb48db21,
      0x96769950b50d88f5, 0xbc143fa4e250eb32, 0xeb194f8e1ae525fe,
      0x92efd1b8d0cf37bf, 0xb7abc627050305ae, 0xe596b7b0c643c71a,
      0x8f7e32ce7bea5c70, 0xb35dbf821ae4f38c, 0xe0352f62a19e306f};
};

template <>
struct cache_holder<ieee754_binary64> {
  using cache_entry_type = wuint::uint128;
  static constexpr int cache_bits = 128;
  static constexpr int min_k = -292;
  static constexpr int max_k = 326;
  static constexpr cache_entry_type cache[] = {
      {0xff77b1fcbebcdc4f, 0x25e8e89c13bb0f7b},
      {0x9faacf3df73609b1, 0x77b191618c54e9ad},
      {0xc795830d75038c1d, 0xd59df5b9ef6a2418},
      {0xf97ae3d0d2446f25, 0x4b0573286b44ad1e},
      {0x9becce62836ac577, 0x4ee367f9430aec33},
      {0xc2e801fb244576d5, 0x229c41f793cda740},
      {0xf3a20279ed56d48a, 0x6b43527578c11110},
      {0x9845418c345644d6, 0x830a13896b78aaaa},
      {0xbe5691ef416bd60c, 0x23cc986bc656d554},
      {0xedec366b11c6cb8f, 0x2cbfbe86b7ec8aa9},
      {0x94b3a202eb1c3f39, 0x7bf7d71432f3d6aa},
      {0xb9e08a83a5e34f07, 0xdaf5ccd93fb0cc54},
      {0xe858ad248f5c22c9, 0xd1b3400f8f9cff69},
      {0x91376c36d99995be, 0x23100809b9c21fa2},
      {0xb58547448ffffb2d, 0xabd40a0c2832a78b},
      {0xe2e69915b3fff9f9, 0x16c90c8f323f516d},
      {0x8dd01fad907ffc3b, 0xae3da7d97f6792e4},
      {0xb1442798f49ffb4a, 0x99cd11cfdf41779d},
      {0xdd95317f31c7fa1d, 0x40405643d711d584},
      {0x8a7d3eef7f1cfc52, 0x482835ea666b2573},
      {0xad1c8eab5ee43b66, 0xda3243650005eed0},
      {0xd863b256369d4a40, 0x90bed43e40076a83},
      {0x873e4f75e2224e68, 0x5a7744a6e804a292},
      {0xa90de3535aaae202, 0x711515d0a205cb37},
      {0xd3515c2831559a83, 0x0d5a5b44ca873e04},
      {0x8412d9991ed58091, 0xe858790afe9486c3},
      {0xa5178fff668ae0b6, 0x626e974dbe39a873},
      {0xce5d73ff402d98e3, 0xfb0a3d212dc81290},
      {0x80fa687f881c7f8e, 0x7ce66634bc9d0b9a},
      {0xa139029f6a239f72, 0x1c1fffc1ebc44e81},
      {0xc987434744ac874e, 0xa327ffb266b56221},
      {0xfbe9141915d7a922, 0x4bf1ff9f0062baa9},
      {0x9d71ac8fada6c9b5, 0x6f773fc3603db4aa},
      {0xc4ce17b399107c22, 0xcb550fb4384d21d4},
      {0xf6019da07f549b2b, 0x7e2a53a146606a49},
      {0x99c102844f94e0fb, 0x2eda7444cbfc426e},
      {0xc0314325637a1939, 0xfa911155fefb5309},
      {0xf03d93eebc589f88, 0x793555ab7eba27cb},
      {0x96267c7535b763b5, 0x4bc1558b2f3458df},
      {0xbbb01b9283253ca2, 0x9eb1aaedfb016f17},
      {0xea9c227723ee8bcb, 0x465e15a979c1cadd},
      {0x92a1958a7675175f, 0x0bfacd89ec191eca},
      {0xb749faed14125d36, 0xcef980ec671f667c},
      {0xe51c79a85916f484, 0x82b7e12780e7401b},
      {0x8f31cc0937ae58d2, 0xd1b2ecb8b0908811},
      {0xb2fe3f0b8599ef07, 0x861fa7e6dcb4aa16},
      {0xdfbdcece67006ac9, 0x67a791e093e1d49b},
      {0x8bd6a141006042bd, 0xe0c8bb2c5c6d24e1},
      {0xaecc49914078536d, 0x58fae9f773886e19},
      {0xda7f5bf590966848, 0xaf39a475506a899f},
      {0x888f99797a5e012d, 0x6d8406c952429604},
      {0xaab37fd7d8f58178, 0xc8e5087ba6d33b84},
      {0xd5605fcdcf32e1d6, 0xfb1e4a9a90880a65},
      {0x855c3be0a17fcd26, 0x5cf2eea09a550680},
      {0xa6b34ad8c9dfc06f, 0xf42faa48c0ea481f},
      {0xd0601d8efc57b08b, 0xf13b94daf124da27},
      {0x823c12795db6ce57, 0x76c53d08d6b70859},
      {0xa2cb1717b52481ed, 0x54768c4b0c64ca6f},
      {0xcb7ddcdda26da268, 0xa9942f5dcf7dfd0a},
      {0xfe5d54150b090b02, 0xd3f93b35435d7c4d},
      {0x9efa548d26e5a6e1, 0xc47bc5014a1a6db0},
      {0xc6b8e9b0709f109a, 0x359ab6419ca1091c},
      {0xf867241c8cc6d4c0, 0xc30163d203c94b63},
      {0x9b407691d7fc44f8, 0x79e0de63425dcf1e},
      {0xc21094364dfb5636, 0x985915fc12f542e5},
      {0xf294b943e17a2bc4, 0x3e6f5b7b17b2939e},
      {0x979cf3ca6cec5b5a, 0xa705992ceecf9c43},
      {0xbd8430bd08277231, 0x50c6ff782a838354},
      {0xece53cec4a314ebd, 0xa4f8bf5635246429},
      {0x940f4613ae5ed136, 0x871b7795e136be9a},
      {0xb913179899f68584, 0x28e2557b59846e40},
      {0xe757dd7ec07426e5, 0x331aeada2fe589d0},
      {0x9096ea6f3848984f, 0x3ff0d2c85def7622},
      {0xb4bca50b065abe63, 0x0fed077a756b53aa},
      {0xe1ebce4dc7f16dfb, 0xd3e8495912c62895},
      {0x8d3360f09cf6e4bd, 0x64712dd7abbbd95d},
      {0xb080392cc4349dec, 0xbd8d794d96aacfb4},
      {0xdca04777f541c567, 0xecf0d7a0fc5583a1},
      {0x89e42caaf9491b60, 0xf41686c49db57245},
      {0xac5d37d5b79b6239, 0x311c2875c522ced6},
      {0xd77485cb25823ac7, 0x7d633293366b828c},
      {0x86a8d39ef77164bc, 0xae5dff9c02033198},
      {0xa8530886b54dbdeb, 0xd9f57f830283fdfd},
      {0xd267caa862a12d66, 0xd072df63c324fd7c},
      {0x8380dea93da4bc60, 0x4247cb9e59f71e6e},
      {0xa46116538d0deb78, 0x52d9be85f074e609},
      {0xcd795be870516656, 0x67902e276c921f8c},
      {0x806bd9714632dff6, 0x00ba1cd8a3db53b7},
      {0xa086cfcd97bf97f3, 0x80e8a40eccd228a5},
      {0xc8a883c0fdaf7df0, 0x6122cd128006b2ce},
      {0xfad2a4b13d1b5d6c, 0x796b805720085f82},
      {0x9cc3a6eec6311a63, 0xcbe3303674053bb1},
      {0xc3f490aa77bd60fc, 0xbedbfc4411068a9d},
      {0xf4f1b4d515acb93b, 0xee92fb5515482d45},
      {0x991711052d8bf3c5, 0x751bdd152d4d1c4b},
      {0xbf5cd54678eef0b6, 0xd262d45a78a0635e},
      {0xef340a98172aace4, 0x86fb897116c87c35},
      {0x9580869f0e7aac0e, 0xd45d35e6ae3d4da1},
      {0xbae0a846d2195712, 0x8974836059cca10a},
      {0xe998d258869facd7, 0x2bd1a438703fc94c},
      {0x91ff83775423cc06, 0x7b6306a34627ddd0},
      {0xb67f6455292cbf08, 0x1a3bc84c17b1d543},
      {0xe41f3d6a7377eeca, 0x20caba5f1d9e4a94},
      {0x8e938662882af53e, 0x547eb47b7282ee9d},
      {0xb23867fb2a35b28d, 0xe99e619a4f23aa44},
      {0xdec681f9f4c31f31, 0x6405fa00e2ec94d5},
      {0x8b3c113c38f9f37e, 0xde83bc408dd3dd05},
      {0xae0b158b4738705e, 0x9624ab50b148d446},
      {0xd98ddaee19068c76, 0x3badd624dd9b0958},
      {0x87f8a8d4cfa417c9, 0xe54ca5d70a80e5d7},
      {0xa9f6d30a038d1dbc, 0x5e9fcf4ccd211f4d},
      {0xd47487cc8470652b, 0x7647c32000696720},
      {0x84c8d4dfd2c63f3b, 0x29ecd9f40041e074},
      {0xa5fb0a17c777cf09, 0xf468107100525891},
      {0xcf79cc9db955c2cc, 0x7182148d4066eeb5},
      {0x81ac1fe293d599bf, 0xc6f14cd848405531},
      {0xa21727db38cb002f, 0xb8ada00e5a506a7d},
      {0xca9cf1d206fdc03b, 0xa6d90811f0e4851d},
      {0xfd442e4688bd304a, 0x908f4a166d1da664},
      {0x9e4a9cec15763e2e, 0x9a598e4e043287ff},
      {0xc5dd44271ad3cdba, 0x40eff1e1853f29fe},
      {0xf7549530e188c128, 0xd12bee59e68ef47d},
      {0x9a94dd3e8cf578b9, 0x82bb74f8301958cf},
      {0xc13a148e3032d6e7, 0xe36a52363c1faf02},
      {0xf18899b1bc3f8ca1, 0xdc44e6c3cb279ac2},
      {0x96f5600f15a7b7e5, 0x29ab103a5ef8c0ba},
      {0xbcb2b812db11a5de, 0x7415d448f6b6f0e8},
      {0xebdf661791d60f56, 0x111b495b3464ad22},
      {0x936b9fcebb25c995, 0xcab10dd900beec35},
      {0xb84687c269ef3bfb, 0x3d5d514f40eea743},
      {0xe65829b3046b0afa, 0x0cb4a5a3112a5113},
      {0x8ff71a0fe2c2e6dc, 0x47f0e785eaba72ac},
      {0xb3f4e093db73a093, 0x59ed216765690f57},
      {0xe0f218b8d25088b8, 0x306869c13ec3532d},
      {0x8c974f7383725573, 0x1e414218c73a13fc},
      {0xafbd2350644eeacf, 0xe5d1929ef90898fb},
      {0xdbac6c247d62a583, 0xdf45f746b74abf3a},
      {0x894bc396ce5da772, 0x6b8bba8c328eb784},
      {0xab9eb47c81f5114f, 0x066ea92f3f326565},
      {0xd686619ba27255a2, 0xc80a537b0efefebe},
      {0x8613fd0145877585, 0xbd06742ce95f5f37},
      {0xa798fc4196e952e7, 0x2c48113823b73705},
      {0xd17f3b51fca3a7a0, 0xf75a15862ca504c6},
      {0x82ef85133de648c4, 0x9a984d73dbe722fc},
      {0xa3ab66580d5fdaf5, 0xc13e60d0d2e0ebbb},
      {0xcc963fee10b7d1b3, 0x318df905079926a9},
      {0xffbbcfe994e5c61f, 0xfdf17746497f7053},
      {0x9fd561f1fd0f9bd3, 0xfeb6ea8bedefa634},
      {0xc7caba6e7c5382c8, 0xfe64a52ee96b8fc1},
      {0xf9bd690a1b68637b, 0x3dfdce7aa3c673b1},
      {0x9c1661a651213e2d, 0x06bea10ca65c084f},
      {0xc31bfa0fe5698db8, 0x486e494fcff30a63},
      {0xf3e2f893dec3f126, 0x5a89dba3c3efccfb},
      {0x986ddb5c6b3a76b7, 0xf89629465a75e01d},
      {0xbe89523386091465, 0xf6bbb397f1135824},
      {0xee2ba6c0678b597f, 0x746aa07ded582e2d},
      {0x94db483840b717ef, 0xa8c2a44eb4571cdd},
      {0xba121a4650e4ddeb, 0x92f34d62616ce414},
      {0xe896a0d7e51e1566, 0x77b020baf9c81d18},
      {0x915e2486ef32cd60, 0x0ace1474dc1d122f},
      {0xb5b5ada8aaff80b8, 0x0d819992132456bb},
      {0xe3231912d5bf60e6, 0x10e1fff697ed6c6a},
      {0x8df5efabc5979c8f, 0xca8d3ffa1ef463c2},
      {0xb1736b96b6fd83b3, 0xbd308ff8a6b17cb3},
      {0xddd0467c64bce4a0, 0xac7cb3f6d05ddbdf},
      {0x8aa22c0dbef60ee4, 0x6bcdf07a423aa96c},
      {0xad4ab7112eb3929d, 0x86c16c98d2c953c7},
      {0xd89d64d57a607744, 0xe871c7bf077ba8b8},
      {0x87625f056c7c4a8b, 0x11471cd764ad4973},
      {0xa93af6c6c79b5d2d, 0xd598e40d3dd89bd0},
      {0xd389b47879823479, 0x4aff1d108d4ec2c4},
      {0x843610cb4bf160cb, 0xcedf722a585139bb},
      {0xa54394fe1eedb8fe, 0xc2974eb4ee658829},
      {0xce947a3da6a9273e, 0x733d226229feea33},
      {0x811ccc668829b887, 0x0806357d5a3f5260},
      {0xa163ff802a3426a8, 0xca07c2dcb0cf26f8},
      {0xc9bcff6034c13052, 0xfc89b393dd02f0b6},
      {0xfc2c3f3841f17c67, 0xbbac2078d443ace3},
      {0x9d9ba7832936edc0, 0xd54b944b84aa4c0e},
      {0xc5029163f384a931, 0x0a9e795e65d4df12},
      {0xf64335bcf065d37d, 0x4d4617b5ff4a16d6},
      {0x99ea0196163fa42e, 0x504bced1bf8e4e46},
      {0xc06481fb9bcf8d39, 0xe45ec2862f71e1d7},
      {0xf07da27a82c37088, 0x5d767327bb4e5a4d},
      {0x964e858c91ba2655, 0x3a6a07f8d510f870},
      {0xbbe226efb628afea, 0x890489f70a55368c},
      {0xeadab0aba3b2dbe5, 0x2b45ac74ccea842f},
      {0x92c8ae6b464fc96f, 0x3b0b8bc90012929e},
      {0xb77ada0617e3bbcb, 0x09ce6ebb40173745},
      {0xe55990879ddcaabd, 0xcc420a6a101d0516},
      {0x8f57fa54c2a9eab6, 0x9fa946824a12232e},
      {0xb32df8e9f3546564, 0x47939822dc96abfa},
      {0xdff9772470297ebd, 0x59787e2b93bc56f8},
      {0x8bfbea76c619ef36, 0x57eb4edb3c55b65b},
      {0xaefae51477a06b03, 0xede622920b6b23f2},
      {0xdab99e59958885c4, 0xe95fab368e45ecee},
      {0x88b402f7fd75539b, 0x11dbcb0218ebb415},
      {0xaae103b5fcd2a881, 0xd652bdc29f26a11a},
      {0xd59944a37c0752a2, 0x4be76d3346f04960},
      {0x857fcae62d8493a5, 0x6f70a4400c562ddc},
      {0xa6dfbd9fb8e5b88e, 0xcb4ccd500f6bb953},
      {0xd097ad07a71f26b2, 0x7e2000a41346a7a8},
      {0x825ecc24c873782f, 0x8ed400668c0c28c9},
      {0xa2f67f2dfa90563b, 0x728900802f0f32fb},
      {0xcbb41ef979346bca, 0x4f2b40a03ad2ffba},
      {0xfea126b7d78186bc, 0xe2f610c84987bfa9},
      {0x9f24b832e6b0f436, 0x0dd9ca7d2df4d7ca},
      {0xc6ede63fa05d3143, 0x91503d1c79720dbc},
      {0xf8a95fcf88747d94, 0x75a44c6397ce912b},
      {0x9b69dbe1b548ce7c, 0xc986afbe3ee11abb},
      {0xc24452da229b021b, 0xfbe85badce996169},
      {0xf2d56790ab41c2a2, 0xfae27299423fb9c4},
      {0x97c560ba6b0919a5, 0xdccd879fc967d41b},
      {0xbdb6b8e905cb600f, 0x5400e987bbc1c921},
      {0xed246723473e3813, 0x290123e9aab23b69},
      {0x9436c0760c86e30b, 0xf9a0b6720aaf6522},
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      {0xe2a0b5dc971f303a, 0x2e44ae64840fd61e},
      {0x8da471a9de737e24, 0x5ceaecfed289e5d3},
      {0xb10d8e1456105dad, 0x7425a83e872c5f48},
      {0xdd50f1996b947518, 0xd12f124e28f7771a},
      {0x8a5296ffe33cc92f, 0x82bd6b70d99aaa70},
      {0xace73cbfdc0bfb7b, 0x636cc64d1001550c},
      {0xd8210befd30efa5a, 0x3c47f7e05401aa4f},
      {0x8714a775e3e95c78, 0x65acfaec34810a72},
      {0xa8d9d1535ce3b396, 0x7f1839a741a14d0e},
      {0xd31045a8341ca07c, 0x1ede48111209a051},
      {0x83ea2b892091e44d, 0x934aed0aab460433},
      {0xa4e4b66b68b65d60, 0xf81da84d56178540},
      {0xce1de40642e3f4b9, 0x36251260ab9d668f},
      {0x80d2ae83e9ce78f3, 0xc1d72b7c6b42601a},
      {0xa1075a24e4421730, 0xb24cf65b8612f820},
      {0xc94930ae1d529cfc, 0xdee033f26797b628},
      {0xfb9b7cd9a4a7443c, 0x169840ef017da3b2},
      {0x9d412e0806e88aa5, 0x8e1f289560ee864f},
      {0xc491798a08a2ad4e, 0xf1a6f2bab92a27e3},
      {0xf5b5d7ec8acb58a2, 0xae10af696774b1dc},
      {0x9991a6f3d6bf1765, 0xacca6da1e0a8ef2a},
      {0xbff610b0cc6edd3f, 0x17fd090a58d32af4},
      {0xeff394dcff8a948e, 0xddfc4b4cef07f5b1},
      {0x95f83d0a1fb69cd9, 0x4abdaf101564f98f},
      {0xbb764c4ca7a4440f, 0x9d6d1ad41abe37f2},
      {0xea53df5fd18d5513, 0x84c86189216dc5ee},
      {0x92746b9be2f8552c, 0x32fd3cf5b4e49bb5},
      {0xb7118682dbb66a77, 0x3fbc8c33221dc2a2},
      {0xe4d5e82392a40515, 0x0fabaf3feaa5334b},
      {0x8f05b1163ba6832d, 0x29cb4d87f2a7400f},
      {0xb2c71d5bca9023f8, 0x743e20e9ef511013},
      {0xdf78e4b2bd342cf6, 0x914da9246b255417},
      {0x8bab8eefb6409c1a, 0x1ad089b6c2f7548f},
      {0xae9672aba3d0c320, 0xa184ac2473b529b2},
      {0xda3c0f568cc4f3e8, 0xc9e5d72d90a2741f},
      {0x8865899617fb1871, 0x7e2fa67c7a658893},
      {0xaa7eebfb9df9de8d, 0xddbb901b98feeab8},
      {0xd51ea6fa85785631, 0x552a74227f3ea566},
      {0x8533285c936b35de, 0xd53a88958f872760},
      {0xa67ff273b8460356, 0x8a892abaf368f138},
      {0xd01fef10a657842c, 0x2d2b7569b0432d86},
      {0x8213f56a67f6b29b, 0x9c3b29620e29fc74},
      {0xa298f2c501f45f42, 0x8349f3ba91b47b90},
      {0xcb3f2f7642717713, 0x241c70a936219a74},
      {0xfe0efb53d30dd4d7, 0xed238cd383aa0111},
      {0x9ec95d1463e8a506, 0xf4363804324a40ab},
      {0xc67bb4597ce2ce48, 0xb143c6053edcd0d6},
      {0xf81aa16fdc1b81da, 0xdd94b7868e94050b},
      {0x9b10a4e5e9913128, 0xca7cf2b4191c8327},
      {0xc1d4ce1f63f57d72, 0xfd1c2f611f63a3f1},
      {0xf24a01a73cf2dccf, 0xbc633b39673c8ced},
      {0x976e41088617ca01, 0xd5be0503e085d814},
      {0xbd49d14aa79dbc82, 0x4b2d8644d8a74e19},
      {0xec9c459d51852ba2, 0xddf8e7d60ed1219f},
      {0x93e1ab8252f33b45, 0xcabb90e5c942b504},
      {0xb8da1662e7b00a17, 0x3d6a751f3b936244},
      {0xe7109bfba19c0c9d, 0x0cc512670a783ad5},
      {0x906a617d450187e2, 0x27fb2b80668b24c6},
      {0xb484f9dc9641e9da, 0xb1f9f660802dedf7},
      {0xe1a63853bbd26451, 0x5e7873f8a0396974},
      {0x8d07e33455637eb2, 0xdb0b487b6423e1e9},
      {0xb049dc016abc5e5f, 0x91ce1a9a3d2cda63},
      {0xdc5c5301c56b75f7, 0x7641a140cc7810fc},
      {0x89b9b3e11b6329ba, 0xa9e904c87fcb0a9e},
      {0xac2820d9623bf429, 0x546345fa9fbdcd45},
      {0xd732290fbacaf133, 0xa97c177947ad4096},
      {0x867f59a9d4bed6c0, 0x49ed8eabcccc485e},
      {0xa81f301449ee8c70, 0x5c68f256bfff5a75},
      {0xd226fc195c6a2f8c, 0x73832eec6fff3112},
      {0x83585d8fd9c25db7, 0xc831fd53c5ff7eac},
      {0xa42e74f3d032f525, 0xba3e7ca8b77f5e56},
      {0xcd3a1230c43fb26f, 0x28ce1bd2e55f35ec},
      {0x80444b5e7aa7cf85, 0x7980d163cf5b81b4},
      {0xa0555e361951c366, 0xd7e105bcc3326220},
      {0xc86ab5c39fa63440, 0x8dd9472bf3fefaa8},
      {0xfa856334878fc150, 0xb14f98f6f0feb952},
      {0x9c935e00d4b9d8d2, 0x6ed1bf9a569f33d4},
      {0xc3b8358109e84f07, 0x0a862f80ec4700c9},
      {0xf4a642e14c6262c8, 0xcd27bb612758c0fb},
      {0x98e7e9cccfbd7dbd, 0x8038d51cb897789d},
      {0xbf21e44003acdd2c, 0xe0470a63e6bd56c4},
      {0xeeea5d5004981478, 0x1858ccfce06cac75},
      {0x95527a5202df0ccb, 0x0f37801e0c43ebc9},
      {0xbaa718e68396cffd, 0xd30560258f54e6bb},
      {0xe950df20247c83fd, 0x47c6b82ef32a206a},
      {0x91d28b7416cdd27e, 0x4cdc331d57fa5442},
      {0xb6472e511c81471d, 0xe0133fe4adf8e953},
      {0xe3d8f9e563a198e5, 0x58180fddd97723a7},
      {0x8e679c2f5e44ff8f, 0x570f09eaa7ea7649},
      {0xb201833b35d63f73, 0x2cd2cc6551e513db},
      {0xde81e40a034bcf4f, 0xf8077f7ea65e58d2},
      {0x8b112e86420f6191, 0xfb04afaf27faf783},
      {0xadd57a27d29339f6, 0x79c5db9af1f9b564},
      {0xd94ad8b1c7380874, 0x18375281ae7822bd},
      {0x87cec76f1c830548, 0x8f2293910d0b15b6},
      {0xa9c2794ae3a3c69a, 0xb2eb3875504ddb23},
      {0xd433179d9c8cb841, 0x5fa60692a46151ec},
      {0x849feec281d7f328, 0xdbc7c41ba6bcd334},
      {0xa5c7ea73224deff3, 0x12b9b522906c0801},
      {0xcf39e50feae16bef, 0xd768226b34870a01},
      {0x81842f29f2cce375, 0xe6a1158300d46641},
      {0xa1e53af46f801c53, 0x60495ae3c1097fd1},
      {0xca5e89b18b602368, 0x385bb19cb14bdfc5},
      {0xfcf62c1dee382c42, 0x46729e03dd9ed7b6},
      {0x9e19db92b4e31ba9, 0x6c07a2c26a8346d2},
      {0xc5a05277621be293, 0xc7098b7305241886},
      {0xf70867153aa2db38, 0xb8cbee4fc66d1ea8}};
};

// Compressed cache for double
struct compressed_cache_detail {
  static constexpr int compression_ratio = 27;
  static constexpr std::size_t compressed_table_size =
      (cache_holder<ieee754_binary64>::max_k -
       cache_holder<ieee754_binary64>::min_k + compression_ratio) /
      compression_ratio;

  struct cache_holder_t {
    wuint::uint128 table[compressed_table_size];
  };
  static constexpr cache_holder_t cache = [] {
    cache_holder_t res{};
    for (std::size_t i = 0; i < compressed_table_size; ++i) {
      res.table[i] =
          cache_holder<ieee754_binary64>::cache[i * compression_ratio];
    }
    return res;
  }();

  struct pow5_holder_t {
    std::uint64_t table[compression_ratio];
  };
  static constexpr pow5_holder_t pow5 = [] {
    pow5_holder_t res{};
    std::uint64_t p = 1;
    for (std::size_t i = 0; i < compression_ratio; ++i) {
      res.table[i] = p;
      p *= 5;
    }
    return res;
  }();
};
}  // namespace detail

////////////////////////////////////////////////////////////////////////////////////////
// Policies.
////////////////////////////////////////////////////////////////////////////////////////

namespace detail {
// Forward declare the implementation class.
template <class Float, class FloatTraits = default_float_traits<Float>>
struct impl;

namespace policy_impl {
// Sign policies.
namespace sign {
struct base {};

struct ignore : base {
  using sign_policy = ignore;
  static constexpr bool return_has_sign = false;

  template <class SignedSignificandBits, class ReturnType>
  static constexpr void handle_sign(SignedSignificandBits,
                                    ReturnType &) noexcept {}
};

struct return_sign : base {
  using sign_policy = return_sign;
  static constexpr bool return_has_sign = true;

  template <class SignedSignificandBits, class ReturnType>
  static constexpr void handle_sign(SignedSignificandBits s,
                                    ReturnType &r) noexcept {
    r.is_negative = s.is_negative();
  }
};
}  // namespace sign

// Trailing zero policies.
namespace trailing_zero {
struct base {};

struct ignore : base {
  using trailing_zero_policy = ignore;
  static constexpr bool report_trailing_zeros = false;

  template <class Impl, class ReturnType>
  static constexpr void on_trailing_zeros(ReturnType &) noexcept {}

  template <class Impl, class ReturnType>
  static constexpr void no_trailing_zeros(ReturnType &) noexcept {}
};

struct remove : base {
  using trailing_zero_policy = remove;
  static constexpr bool report_trailing_zeros = false;

  template <class Impl, class ReturnType>
  JKJ_FORCEINLINE static constexpr void on_trailing_zeros(
      ReturnType &r) noexcept {
    r.exponent += Impl::remove_trailing_zeros(r.significand);
  }

  template <class Impl, class ReturnType>
  static constexpr void no_trailing_zeros(ReturnType &) noexcept {}
};

struct report : base {
  using trailing_zero_policy = report;
  static constexpr bool report_trailing_zeros = true;

  template <class Impl, class ReturnType>
  static constexpr void on_trailing_zeros(ReturnType &r) noexcept {
    r.may_have_trailing_zeros = true;
  }

  template <class Impl, class ReturnType>
  static constexpr void no_trailing_zeros(ReturnType &r) noexcept {
    r.may_have_trailing_zeros = false;
  }
};
}  // namespace trailing_zero

// Decimal-to-binary rounding mode policies.
namespace decimal_to_binary_rounding {
struct base {};

enum class tag_t { to_nearest, left_closed_directed, right_closed_directed };
namespace interval_type {
struct symmetric_boundary {
  static constexpr bool is_symmetric = true;
  bool is_closed;
  constexpr bool include_left_endpoint() const noexcept { return is_closed; }
  constexpr bool include_right_endpoint() const noexcept { return is_closed; }
};
struct asymmetric_boundary {
  static constexpr bool is_symmetric = false;
  bool is_left_closed;
  constexpr bool include_left_endpoint() const noexcept {
    return is_left_closed;
  }
  constexpr bool include_right_endpoint() const noexcept {
    return !is_left_closed;
  }
};
struct closed {
  static constexpr bool is_symmetric = true;
  static constexpr bool include_left_endpoint() noexcept { return true; }
  static constexpr bool include_right_endpoint() noexcept { return true; }
};
struct open {
  static constexpr bool is_symmetric = true;
  static constexpr bool include_left_endpoint() noexcept { return false; }
  static constexpr bool include_right_endpoint() noexcept { return false; }
};
struct left_closed_right_open {
  static constexpr bool is_symmetric = false;
  static constexpr bool include_left_endpoint() noexcept { return true; }
  static constexpr bool include_right_endpoint() noexcept { return false; }
};
struct right_closed_left_open {
  static constexpr bool is_symmetric = false;
  static constexpr bool include_left_endpoint() noexcept { return false; }
  static constexpr bool include_right_endpoint() noexcept { return true; }
};
}  // namespace interval_type

struct nearest_to_even : base {
  using decimal_to_binary_rounding_policy = nearest_to_even;
  static constexpr auto tag = tag_t::to_nearest;
  using normal_interval_type = interval_type::symmetric_boundary;
  using shorter_interval_type = interval_type::closed;

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits,
                                       Func &&f) noexcept {
    return f(nearest_to_even{});
  }

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_normal_interval_case(
      SignedSignificandBits s, Func &&f) noexcept {
    return f(s.has_even_significand_bits());
  }
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_shorter_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
};
struct nearest_to_odd : base {
  using decimal_to_binary_rounding_policy = nearest_to_odd;
  static constexpr auto tag = tag_t::to_nearest;
  using normal_interval_type = interval_type::symmetric_boundary;
  using shorter_interval_type = interval_type::open;

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits,
                                       Func &&f) noexcept {
    return f(nearest_to_odd{});
  }

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_normal_interval_case(
      SignedSignificandBits s, Func &&f) noexcept {
    return f(!s.has_even_significand_bits());
  }
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_shorter_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
};
struct nearest_toward_plus_infinity : base {
  using decimal_to_binary_rounding_policy = nearest_toward_plus_infinity;
  static constexpr auto tag = tag_t::to_nearest;
  using normal_interval_type = interval_type::asymmetric_boundary;
  using shorter_interval_type = interval_type::asymmetric_boundary;

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits,
                                       Func &&f) noexcept {
    return f(nearest_toward_plus_infinity{});
  }

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_normal_interval_case(
      SignedSignificandBits s, Func &&f) noexcept {
    return f(!s.is_negative());
  }
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_shorter_interval_case(
      SignedSignificandBits s, Func &&f) noexcept {
    return f(!s.is_negative());
  }
};
struct nearest_toward_minus_infinity : base {
  using decimal_to_binary_rounding_policy = nearest_toward_minus_infinity;
  static constexpr auto tag = tag_t::to_nearest;
  using normal_interval_type = interval_type::asymmetric_boundary;
  using shorter_interval_type = interval_type::asymmetric_boundary;

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits,
                                       Func &&f) noexcept {
    return f(nearest_toward_minus_infinity{});
  }

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_normal_interval_case(
      SignedSignificandBits s, Func &&f) noexcept {
    return f(s.is_negative());
  }
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_shorter_interval_case(
      SignedSignificandBits s, Func &&f) noexcept {
    return f(s.is_negative());
  }
};
struct nearest_toward_zero : base {
  using decimal_to_binary_rounding_policy = nearest_toward_zero;
  static constexpr auto tag = tag_t::to_nearest;
  using normal_interval_type = interval_type::right_closed_left_open;
  using shorter_interval_type = interval_type::right_closed_left_open;

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits,
                                       Func &&f) noexcept {
    return f(nearest_toward_zero{});
  }

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_normal_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_shorter_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
};
struct nearest_away_from_zero : base {
  using decimal_to_binary_rounding_policy = nearest_away_from_zero;
  static constexpr auto tag = tag_t::to_nearest;
  using normal_interval_type = interval_type::left_closed_right_open;
  using shorter_interval_type = interval_type::left_closed_right_open;

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits,
                                       Func &&f) noexcept {
    return f(nearest_away_from_zero{});
  }

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_normal_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_shorter_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
};

namespace detail {
struct nearest_always_closed {
  static constexpr auto tag = tag_t::to_nearest;
  using normal_interval_type = interval_type::closed;
  using shorter_interval_type = interval_type::closed;

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_normal_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_shorter_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
};
struct nearest_always_open {
  static constexpr auto tag = tag_t::to_nearest;
  using normal_interval_type = interval_type::open;
  using shorter_interval_type = interval_type::open;

  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_normal_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static constexpr auto invoke_shorter_interval_case(
      SignedSignificandBits, Func &&f) noexcept {
    return f();
  }
};
}  // namespace detail

struct nearest_to_even_static_boundary : base {
  using decimal_to_binary_rounding_policy = nearest_to_even_static_boundary;
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                       Func &&f) noexcept {
    if (s.has_even_significand_bits()) {
      return f(detail::nearest_always_closed{});
    }
    else {
      return f(detail::nearest_always_open{});
    }
  }
};
struct nearest_to_odd_static_boundary : base {
  using decimal_to_binary_rounding_policy = nearest_to_odd_static_boundary;
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                       Func &&f) noexcept {
    if (s.has_even_significand_bits()) {
      return f(detail::nearest_always_open{});
    }
    else {
      return f(detail::nearest_always_closed{});
    }
  }
};
struct nearest_toward_plus_infinity_static_boundary : base {
  using decimal_to_binary_rounding_policy =
      nearest_toward_plus_infinity_static_boundary;
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                       Func &&f) noexcept {
    if (s.is_negative()) {
      return f(nearest_toward_zero{});
    }
    else {
      return f(nearest_away_from_zero{});
    }
  }
};
struct nearest_toward_minus_infinity_static_boundary : base {
  using decimal_to_binary_rounding_policy =
      nearest_toward_minus_infinity_static_boundary;
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                       Func &&f) noexcept {
    if (s.is_negative()) {
      return f(nearest_away_from_zero{});
    }
    else {
      return f(nearest_toward_zero{});
    }
  }
};

namespace detail {
struct left_closed_directed {
  static constexpr auto tag = tag_t::left_closed_directed;
};
struct right_closed_directed {
  static constexpr auto tag = tag_t::right_closed_directed;
};
}  // namespace detail

struct toward_plus_infinity : base {
  using decimal_to_binary_rounding_policy = toward_plus_infinity;
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                       Func &&f) noexcept {
    if (s.is_negative()) {
      return f(detail::left_closed_directed{});
    }
    else {
      return f(detail::right_closed_directed{});
    }
  }
};
struct toward_minus_infinity : base {
  using decimal_to_binary_rounding_policy = toward_minus_infinity;
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                       Func &&f) noexcept {
    if (s.is_negative()) {
      return f(detail::right_closed_directed{});
    }
    else {
      return f(detail::left_closed_directed{});
    }
  }
};
struct toward_zero : base {
  using decimal_to_binary_rounding_policy = toward_zero;
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits,
                                       Func &&f) noexcept {
    return f(detail::left_closed_directed{});
  }
};
struct away_from_zero : base {
  using decimal_to_binary_rounding_policy = away_from_zero;
  template <class SignedSignificandBits, class Func>
  JKJ_FORCEINLINE static auto delegate(SignedSignificandBits,
                                       Func &&f) noexcept {
    return f(detail::right_closed_directed{});
  }
};
}  // namespace decimal_to_binary_rounding

// Binary-to-decimal rounding policies.
// (Always assumes nearest rounding modes.)
namespace binary_to_decimal_rounding {
struct base {};

enum class tag_t { do_not_care, to_even, to_odd, away_from_zero, toward_zero };

struct do_not_care : base {
  using binary_to_decimal_rounding_policy = do_not_care;
  static constexpr auto tag = tag_t::do_not_care;

  template <class ReturnType>
  static constexpr bool prefer_round_down(ReturnType const &) noexcept {
    return false;
  }
};

struct to_even : base {
  using binary_to_decimal_rounding_policy = to_even;
  static constexpr auto tag = tag_t::to_even;

  template <class ReturnType>
  static constexpr bool prefer_round_down(ReturnType const &r) noexcept {
    return r.significand % 2 != 0;
  }
};

struct to_odd : base {
  using binary_to_decimal_rounding_policy = to_odd;
  static constexpr auto tag = tag_t::to_odd;

  template <class ReturnType>
  static constexpr bool prefer_round_down(ReturnType const &r) noexcept {
    return r.significand % 2 == 0;
  }
};

struct away_from_zero : base {
  using binary_to_decimal_rounding_policy = away_from_zero;
  static constexpr auto tag = tag_t::away_from_zero;

  template <class ReturnType>
  static constexpr bool prefer_round_down(ReturnType const &) noexcept {
    return false;
  }
};

struct toward_zero : base {
  using binary_to_decimal_rounding_policy = toward_zero;
  static constexpr auto tag = tag_t::toward_zero;

  template <class ReturnType>
  static constexpr bool prefer_round_down(ReturnType const &) noexcept {
    return true;
  }
};
}  // namespace binary_to_decimal_rounding

// Cache policies.
namespace cache {
struct base {};

struct full : base {
  using cache_policy = full;
  template <class FloatFormat>
  static constexpr typename cache_holder<FloatFormat>::cache_entry_type
  get_cache(int k) noexcept {
    assert(k >= cache_holder<FloatFormat>::min_k &&
           k <= cache_holder<FloatFormat>::max_k);
    return cache_holder<FloatFormat>::cache[std::size_t(
        k - cache_holder<FloatFormat>::min_k)];
  }
};

struct compact : base {
  using cache_policy = compact;
  template <class FloatFormat>
  static constexpr typename cache_holder<FloatFormat>::cache_entry_type
  get_cache(int k) noexcept {
    assert(k >= cache_holder<FloatFormat>::min_k &&
           k <= cache_holder<FloatFormat>::max_k);

    if constexpr (std::is_same_v<FloatFormat, ieee754_binary64>) {
      // Compute the base index.
      auto const cache_index =
          int(std::uint32_t(k - cache_holder<FloatFormat>::min_k) /
              compressed_cache_detail::compression_ratio);
      auto const kb = cache_index * compressed_cache_detail::compression_ratio +
                      cache_holder<FloatFormat>::min_k;
      auto const offset = k - kb;

      // Get the base cache.
      auto const base_cache = compressed_cache_detail::cache.table[cache_index];

      if (offset == 0) {
        return base_cache;
      }
      else {
        // Compute the required amount of bit-shift.
        auto const alpha = log::floor_log2_pow10(kb + offset) -
                           log::floor_log2_pow10(kb) - offset;
        assert(alpha > 0 && alpha < 64);

        // Try to recover the real cache.
        auto const pow5 = compressed_cache_detail::pow5.table[offset];
        auto recovered_cache = wuint::umul128(base_cache.high(), pow5);
        auto const middle_low = wuint::umul128(base_cache.low(), pow5);

        recovered_cache += middle_low.high();

        auto const high_to_middle = recovered_cache.high() << (64 - alpha);
        auto const middle_to_low = recovered_cache.low() << (64 - alpha);

        recovered_cache =
            wuint::uint128{(recovered_cache.low() >> alpha) | high_to_middle,
                           ((middle_low.low() >> alpha) | middle_to_low)};

        assert(recovered_cache.low() + 1 != 0);
        recovered_cache = {recovered_cache.high(), recovered_cache.low() + 1};

        return recovered_cache;
      }
    }
    else {
      // Just use the full cache for anything other than binary64
      return cache_holder<FloatFormat>::cache[std::size_t(
          k - cache_holder<FloatFormat>::min_k)];
    }
  }
};
}  // namespace cache
}  // namespace policy_impl
}  // namespace detail

namespace policy {
namespace sign {
inline constexpr auto ignore = detail::policy_impl::sign::ignore{};
inline constexpr auto return_sign = detail::policy_impl::sign::return_sign{};
}  // namespace sign

namespace trailing_zero {
inline constexpr auto ignore = detail::policy_impl::trailing_zero::ignore{};
inline constexpr auto remove = detail::policy_impl::trailing_zero::remove{};
inline constexpr auto report = detail::policy_impl::trailing_zero::report{};
}  // namespace trailing_zero

namespace decimal_to_binary_rounding {
inline constexpr auto nearest_to_even =
    detail::policy_impl::decimal_to_binary_rounding::nearest_to_even{};
inline constexpr auto nearest_to_odd =
    detail::policy_impl::decimal_to_binary_rounding::nearest_to_odd{};
inline constexpr auto nearest_toward_plus_infinity = detail::policy_impl::
    decimal_to_binary_rounding::nearest_toward_plus_infinity{};
inline constexpr auto nearest_toward_minus_infinity = detail::policy_impl::
    decimal_to_binary_rounding::nearest_toward_minus_infinity{};
inline constexpr auto nearest_toward_zero =
    detail::policy_impl::decimal_to_binary_rounding::nearest_toward_zero{};
inline constexpr auto nearest_away_from_zero =
    detail::policy_impl::decimal_to_binary_rounding::nearest_away_from_zero{};

inline constexpr auto nearest_to_even_static_boundary = detail::policy_impl::
    decimal_to_binary_rounding::nearest_to_even_static_boundary{};
inline constexpr auto nearest_to_odd_static_boundary = detail::policy_impl::
    decimal_to_binary_rounding::nearest_to_odd_static_boundary{};
inline constexpr auto nearest_toward_plus_infinity_static_boundary =
    detail::policy_impl::decimal_to_binary_rounding::
        nearest_toward_plus_infinity_static_boundary{};
inline constexpr auto nearest_toward_minus_infinity_static_boundary =
    detail::policy_impl::decimal_to_binary_rounding::
        nearest_toward_minus_infinity_static_boundary{};

inline constexpr auto toward_plus_infinity =
    detail::policy_impl::decimal_to_binary_rounding::toward_plus_infinity{};
inline constexpr auto toward_minus_infinity =
    detail::policy_impl::decimal_to_binary_rounding::toward_minus_infinity{};
inline constexpr auto toward_zero =
    detail::policy_impl::decimal_to_binary_rounding::toward_zero{};
inline constexpr auto away_from_zero =
    detail::policy_impl::decimal_to_binary_rounding::away_from_zero{};
}  // namespace decimal_to_binary_rounding

namespace binary_to_decimal_rounding {
inline constexpr auto do_not_care =
    detail::policy_impl::binary_to_decimal_rounding::do_not_care{};
inline constexpr auto to_even =
    detail::policy_impl::binary_to_decimal_rounding::to_even{};
inline constexpr auto to_odd =
    detail::policy_impl::binary_to_decimal_rounding::to_odd{};
inline constexpr auto away_from_zero =
    detail::policy_impl::binary_to_decimal_rounding::away_from_zero{};
inline constexpr auto toward_zero =
    detail::policy_impl::binary_to_decimal_rounding::toward_zero{};
}  // namespace binary_to_decimal_rounding

namespace cache {
inline constexpr auto full = detail::policy_impl::cache::full{};
inline constexpr auto compact = detail::policy_impl::cache::compact{};
}  // namespace cache
}  // namespace policy

namespace detail {
////////////////////////////////////////////////////////////////////////////////////////
// The main algorithm.
////////////////////////////////////////////////////////////////////////////////////////

template <class Float, class FloatTraits>
struct impl : private FloatTraits, private FloatTraits::format {
  using format = typename FloatTraits::format;
  using carrier_uint = typename FloatTraits::carrier_uint;

  using FloatTraits::carrier_bits;
  using format::decimal_digits;
  using format::exponent_bias;
  using format::max_exponent;
  using format::min_exponent;
  using format::significand_bits;

  static constexpr int kappa = std::is_same_v<format, ieee754_binary32> ? 1 : 2;
  static_assert(kappa >= 1);
  static_assert(carrier_bits >=
                significand_bits + 2 + log::floor_log2_pow10(kappa + 1));

  static constexpr int min_k = [] {
    constexpr auto a = -log::floor_log10_pow2_minus_log10_4_over_3(
        int(max_exponent - significand_bits));
    constexpr auto b =
        -log::floor_log10_pow2(int(max_exponent - significand_bits)) + kappa;
    return a < b ? a : b;
  }();
  static_assert(min_k >= cache_holder<format>::min_k);

  static constexpr int max_k = [] {
    // We do invoke shorter_interval_case for exponent == min_exponent case,
    // so we should not add 1 here.
    constexpr auto a = -log::floor_log10_pow2_minus_log10_4_over_3(
        int(min_exponent - significand_bits /*+ 1*/));
    constexpr auto b =
        -log::floor_log10_pow2(int(min_exponent - significand_bits)) + kappa;
    return a > b ? a : b;
  }();
  static_assert(max_k <= cache_holder<format>::max_k);

  using cache_entry_type = typename cache_holder<format>::cache_entry_type;
  static constexpr auto cache_bits = cache_holder<format>::cache_bits;

  static constexpr int case_shorter_interval_left_endpoint_lower_threshold = 2;
  static constexpr int case_shorter_interval_left_endpoint_upper_threshold =
      2 +
      log::floor_log2(
          compute_power<count_factors<5>(
                            (carrier_uint(1) << (significand_bits + 2)) - 1) +
                        1>(10) /
          3);

  static constexpr int case_shorter_interval_right_endpoint_lower_threshold = 0;
  static constexpr int case_shorter_interval_right_endpoint_upper_threshold =
      2 +
      log::floor_log2(
          compute_power<count_factors<5>(
                            (carrier_uint(1) << (significand_bits + 1)) + 1) +
                        1>(10) /
          3);

  static constexpr int shorter_interval_tie_lower_threshold =
      -log::floor_log5_pow2_minus_log5_3(significand_bits + 4) - 2 -
      significand_bits;
  static constexpr int shorter_interval_tie_upper_threshold =
      -log::floor_log5_pow2(significand_bits + 2) - 2 - significand_bits;

  struct compute_mul_result {
    carrier_uint result;
    bool is_integer;
  };
  struct compute_mul_parity_result {
    bool parity;
    bool is_integer;
  };

  //// The main algorithm assumes the input is a normal/subnormal finite number

  template <class ReturnType, class IntervalType, class TrailingZeroPolicy,
            class BinaryToDecimalRoundingPolicy, class CachePolicy,
            class... AdditionalArgs>
  JKJ_SAFEBUFFERS static ReturnType compute_nearest_normal(
      carrier_uint const two_fc, int const exponent,
      AdditionalArgs... additional_args) noexcept {
    //////////////////////////////////////////////////////////////////////
    // Step 1: Schubfach multiplier calculation
    //////////////////////////////////////////////////////////////////////

    ReturnType ret_value;
    IntervalType interval_type{additional_args...};

    // Compute k and beta.
    int const minus_k = log::floor_log10_pow2(exponent) - kappa;
    auto const cache = CachePolicy::template get_cache<format>(-minus_k);
    int const beta = exponent + log::floor_log2_pow10(-minus_k);

    // Compute zi and deltai.
    // 10^kappa <= deltai < 10^(kappa + 1)
    auto const deltai = compute_delta(cache, beta);
    // For the case of binary32, the result of integer check is not correct for
    // 29711844 * 2^-82
    // = 6.1442653300000000008655037797566933477355632930994033813476... *
    // 10^-18 and 29711844 * 2^-81
    // = 1.2288530660000000001731007559513386695471126586198806762695... *
    // 10^-17, and they are the unique counterexamples. However, since 29711844
    // is even, this does not cause any problem for the endpoints calculations;
    // it can only cause a problem when we need to perform integer check for the
    // center. Fortunately, with these inputs, that branch is never executed, so
    // we are fine.
    auto const [zi, is_z_integer] = compute_mul((two_fc | 1) << beta, cache);

    //////////////////////////////////////////////////////////////////////
    // Step 2: Try larger divisor; remove trailing zeros if necessary
    //////////////////////////////////////////////////////////////////////

    constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));
    constexpr auto small_divisor = compute_power<kappa>(std::uint32_t(10));

    // Using an upper bound on zi, we might be able to optimize the division
    // better than the compiler; we are computing zi / big_divisor here.
    ret_value.significand = div::divide_by_pow10<
        kappa + 1, carrier_uint,
        (carrier_uint(1) << (significand_bits + 1)) * big_divisor - 1>(zi);
    auto r = std::uint32_t(zi - big_divisor * ret_value.significand);

    if (r < deltai) {
      // Exclude the right endpoint if necessary.
      if (r == 0 && (is_z_integer & !interval_type.include_right_endpoint())) {
        if constexpr (BinaryToDecimalRoundingPolicy::tag ==
                      policy_impl::binary_to_decimal_rounding::tag_t::
                          do_not_care) {
          ret_value.significand *= 10;
          ret_value.exponent = minus_k + kappa;
          --ret_value.significand;
          TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
          return ret_value;
        }
        else {
          --ret_value.significand;
          r = big_divisor;
          goto small_divisor_case_label;
        }
      }
    }
    else if (r > deltai) {
      goto small_divisor_case_label;
    }
    else {
      // r == deltai; compare fractional parts.
      auto const [xi_parity, x_is_integer] =
          compute_mul_parity(two_fc - 1, cache, beta);

      if (!(xi_parity |
            (x_is_integer & interval_type.include_left_endpoint()))) {
        goto small_divisor_case_label;
      }
    }
    ret_value.exponent = minus_k + kappa + 1;

    // We may need to remove trailing zeros.
    TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
    return ret_value;

    //////////////////////////////////////////////////////////////////////
    // Step 3: Find the significand with the smaller divisor
    //////////////////////////////////////////////////////////////////////

  small_divisor_case_label:
    TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
    ret_value.significand *= 10;
    ret_value.exponent = minus_k + kappa;

    if constexpr (BinaryToDecimalRoundingPolicy::tag ==
                  policy_impl::binary_to_decimal_rounding::tag_t::do_not_care) {
      // Normally, we want to compute
      // ret_value.significand += r / small_divisor
      // and return, but we need to take care of the case that the resulting
      // value is exactly the right endpoint, while that is not included in the
      // interval.
      if (!interval_type.include_right_endpoint()) {
        // Is r divisible by 10^kappa?
        if (is_z_integer &&
            div::check_divisibility_and_divide_by_pow10<kappa>(r)) {
          // This should be in the interval.
          ret_value.significand += r - 1;
        }
        else {
          ret_value.significand += r;
        }
      }
      else {
        ret_value.significand += div::small_division_by_pow10<kappa>(r);
      }
    }
    else {
      auto dist = r - (deltai / 2) + (small_divisor / 2);
      bool const approx_y_parity = ((dist ^ (small_divisor / 2)) & 1) != 0;

      // Is dist divisible by 10^kappa?
      bool const divisible_by_small_divisor =
          div::check_divisibility_and_divide_by_pow10<kappa>(dist);

      // Add dist / 10^kappa to the significand.
      ret_value.significand += dist;

      if (divisible_by_small_divisor) {
        // Check z^(f) >= epsilon^(f).
        // We have either yi == zi - epsiloni or yi == (zi - epsiloni) - 1,
        // where yi == zi - epsiloni if and only if z^(f) >= epsilon^(f).
        // Since there are only 2 possibilities, we only need to care about the
        // parity. Also, zi and r should have the same parity since the divisor
        // is an even number.
        auto const [yi_parity, is_y_integer] =
            compute_mul_parity(two_fc, cache, beta);
        if (yi_parity != approx_y_parity) {
          --ret_value.significand;
        }
        else {
          // If z^(f) >= epsilon^(f), we might have a tie
          // when z^(f) == epsilon^(f), or equivalently, when y is an integer.
          // For tie-to-up case, we can just choose the upper one.
          if (BinaryToDecimalRoundingPolicy::prefer_round_down(ret_value) &
              is_y_integer) {
            --ret_value.significand;
          }
        }
      }
    }
    return ret_value;
  }

  template <class ReturnType, class IntervalType, class TrailingZeroPolicy,
            class BinaryToDecimalRoundingPolicy, class CachePolicy,
            class... AdditionalArgs>
  JKJ_SAFEBUFFERS static ReturnType compute_nearest_shorter(
      int const exponent, AdditionalArgs... additional_args) noexcept {
    ReturnType ret_value;
    IntervalType interval_type{additional_args...};

    // Compute k and beta.
    int const minus_k = log::floor_log10_pow2_minus_log10_4_over_3(exponent);
    int const beta = exponent + log::floor_log2_pow10(-minus_k);

    // Compute xi and zi.
    auto const cache = CachePolicy::template get_cache<format>(-minus_k);

    auto xi = compute_left_endpoint_for_shorter_interval_case(cache, beta);
    auto zi = compute_right_endpoint_for_shorter_interval_case(cache, beta);

    // If we don't accept the right endpoint and
    // if the right endpoint is an integer, decrease it.
    if (!interval_type.include_right_endpoint() &&
        is_right_endpoint_integer_shorter_interval(exponent)) {
      --zi;
    }
    // If we don't accept the left endpoint or
    // if the left endpoint is not an integer, increase it.
    if (!interval_type.include_left_endpoint() ||
        !is_left_endpoint_integer_shorter_interval(exponent)) {
      ++xi;
    }

    // Try bigger divisor.
    ret_value.significand = zi / 10;

    // If succeed, remove trailing zeros if necessary and return.
    if (ret_value.significand * 10 >= xi) {
      ret_value.exponent = minus_k + 1;
      TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
      return ret_value;
    }

    // Otherwise, compute the round-up of y.
    TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
    ret_value.significand =
        compute_round_up_for_shorter_interval_case(cache, beta);
    ret_value.exponent = minus_k;

    // When tie occurs, choose one of them according to the rule.
    if (BinaryToDecimalRoundingPolicy::prefer_round_down(ret_value) &&
        exponent >= shorter_interval_tie_lower_threshold &&
        exponent <= shorter_interval_tie_upper_threshold) {
      --ret_value.significand;
    }
    else if (ret_value.significand < xi) {
      ++ret_value.significand;
    }
    return ret_value;
  }

  template <class ReturnType, class TrailingZeroPolicy, class CachePolicy>
  JKJ_SAFEBUFFERS static ReturnType compute_left_closed_directed(
      carrier_uint const two_fc, int exponent) noexcept {
    //////////////////////////////////////////////////////////////////////
    // Step 1: Schubfach multiplier calculation
    //////////////////////////////////////////////////////////////////////

    ReturnType ret_value;

    // Compute k and beta.
    int const minus_k = log::floor_log10_pow2(exponent) - kappa;
    auto const cache = CachePolicy::template get_cache<format>(-minus_k);
    int const beta = exponent + log::floor_log2_pow10(-minus_k);

    // Compute xi and deltai.
    // 10^kappa <= deltai < 10^(kappa + 1)
    auto const deltai = compute_delta(cache, beta);
    auto [xi, is_x_integer] = compute_mul(two_fc << beta, cache);

    // Deal with the unique exceptional cases
    // 29711844 * 2^-82
    // = 6.1442653300000000008655037797566933477355632930994033813476... *
    // 10^-18 and 29711844 * 2^-81
    // = 1.2288530660000000001731007559513386695471126586198806762695... *
    // 10^-17 for binary32.
    if constexpr (std::is_same_v<format, ieee754_binary32>) {
      if (exponent <= -80) {
        is_x_integer = false;
      }
    }

    if (!is_x_integer) {
      ++xi;
    }

    //////////////////////////////////////////////////////////////////////
    // Step 2: Try larger divisor; remove trailing zeros if necessary
    //////////////////////////////////////////////////////////////////////

    constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));

    // Using an upper bound on xi, we might be able to optimize the division
    // better than the compiler; we are computing xi / big_divisor here.
    ret_value.significand = div::divide_by_pow10<
        kappa + 1, carrier_uint,
        (carrier_uint(1) << (significand_bits + 1)) * big_divisor - 1>(xi);
    auto r = std::uint32_t(xi - big_divisor * ret_value.significand);

    if (r != 0) {
      ++ret_value.significand;
      r = big_divisor - r;
    }

    if (r > deltai) {
      goto small_divisor_case_label;
    }
    else if (r == deltai) {
      // Compare the fractional parts.
      // This branch is never taken for the exceptional cases
      // 2f_c = 29711482, e = -81
      // (6.1442649164096937243516663440523473127541365101933479309082... *
      // 10^-18) and 2f_c = 29711482, e = -80
      // (1.2288529832819387448703332688104694625508273020386695861816... *
      // 10^-17).
      auto const [zi_parity, is_z_integer] =
          compute_mul_parity(two_fc + 2, cache, beta);
      if (zi_parity || is_z_integer) {
        goto small_divisor_case_label;
      }
    }

    // The ceiling is inside, so we are done.
    ret_value.exponent = minus_k + kappa + 1;
    TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
    return ret_value;

    //////////////////////////////////////////////////////////////////////
    // Step 3: Find the significand with the smaller divisor
    //////////////////////////////////////////////////////////////////////

  small_divisor_case_label:
    ret_value.significand *= 10;
    ret_value.significand -= div::small_division_by_pow10<kappa>(r);
    ret_value.exponent = minus_k + kappa;
    TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
    return ret_value;
  }

  template <class ReturnType, class TrailingZeroPolicy, class CachePolicy>
  JKJ_SAFEBUFFERS static ReturnType compute_right_closed_directed(
      carrier_uint const two_fc, int const exponent,
      bool shorter_interval) noexcept {
    //////////////////////////////////////////////////////////////////////
    // Step 1: Schubfach multiplier calculation
    //////////////////////////////////////////////////////////////////////

    ReturnType ret_value;

    // Compute k and beta.
    int const minus_k =
        log::floor_log10_pow2(exponent - (shorter_interval ? 1 : 0)) - kappa;
    auto const cache = CachePolicy::template get_cache<format>(-minus_k);
    int const beta = exponent + log::floor_log2_pow10(-minus_k);

    // Compute zi and deltai.
    // 10^kappa <= deltai < 10^(kappa + 1)
    auto const deltai = shorter_interval ? compute_delta(cache, beta - 1)
                                         : compute_delta(cache, beta);
    carrier_uint const zi = compute_mul(two_fc << beta, cache).result;

    //////////////////////////////////////////////////////////////////////
    // Step 2: Try larger divisor; remove trailing zeros if necessary
    //////////////////////////////////////////////////////////////////////

    constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));

    // Using an upper bound on zi, we might be able to optimize the division
    // better than the compiler; we are computing zi / big_divisor here.
    ret_value.significand = div::divide_by_pow10<
        kappa + 1, carrier_uint,
        (carrier_uint(1) << (significand_bits + 1)) * big_divisor - 1>(zi);
    auto const r = std::uint32_t(zi - big_divisor * ret_value.significand);

    if (r > deltai) {
      goto small_divisor_case_label;
    }
    else if (r == deltai) {
      // Compare the fractional parts.
      if (!compute_mul_parity(two_fc - (shorter_interval ? 1 : 2), cache, beta)
               .parity) {
        goto small_divisor_case_label;
      }
    }

    // The floor is inside, so we are done.
    ret_value.exponent = minus_k + kappa + 1;
    TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
    return ret_value;

    //////////////////////////////////////////////////////////////////////
    // Step 3: Find the significand with the small divisor
    //////////////////////////////////////////////////////////////////////

  small_divisor_case_label:
    ret_value.significand *= 10;
    ret_value.significand += div::small_division_by_pow10<kappa>(r);
    ret_value.exponent = minus_k + kappa;
    TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
    return ret_value;
  }

  // Remove trailing zeros from n and return the number of zeros removed.
  JKJ_FORCEINLINE static int remove_trailing_zeros(carrier_uint &n) noexcept {
    assert(n != 0);

    if constexpr (std::is_same_v<format, ieee754_binary32>) {
      constexpr auto mod_inv_5 = std::uint32_t(0xcccc'cccd);
      constexpr auto mod_inv_25 = mod_inv_5 * mod_inv_5;

      int s = 0;
      while (true) {
        auto q = bits::rotr(n * mod_inv_25, 2);
        if (q <= std::numeric_limits<std::uint32_t>::max() / 100) {
          n = q;
          s += 2;
        }
        else {
          break;
        }
      }
      auto q = bits::rotr(n * mod_inv_5, 1);
      if (q <= std::numeric_limits<std::uint32_t>::max() / 10) {
        n = q;
        s |= 1;
      }

      return s;
    }
    else {
      static_assert(std::is_same_v<format, ieee754_binary64>);

      // Divide by 10^8 and reduce to 32-bits if divisible.
      // Since ret_value.significand <= (2^53 * 1000 - 1) / 1000 < 10^16,
      // n is at most of 16 digits.

      // This magic number is ceil(2^90 / 10^8).
      constexpr auto magic_number = std::uint64_t(12379400392853802749ull);
      auto nm = wuint::umul128(n, magic_number);

      // Is n is divisible by 10^8?
      if ((nm.high() & ((std::uint64_t(1) << (90 - 64)) - 1)) == 0 &&
          nm.low() < magic_number) {
        // If yes, work with the quotient.
        auto n32 = std::uint32_t(nm.high() >> (90 - 64));

        constexpr auto mod_inv_5 = std::uint32_t(0xcccc'cccd);
        constexpr auto mod_inv_25 = mod_inv_5 * mod_inv_5;

        int s = 8;
        while (true) {
          auto q = bits::rotr(n32 * mod_inv_25, 2);
          if (q <= std::numeric_limits<std::uint32_t>::max() / 100) {
            n32 = q;
            s += 2;
          }
          else {
            break;
          }
        }
        auto q = bits::rotr(n32 * mod_inv_5, 1);
        if (q <= std::numeric_limits<std::uint32_t>::max() / 10) {
          n32 = q;
          s |= 1;
        }

        n = n32;
        return s;
      }

      // If n is not divisible by 10^8, work with n itself.
      constexpr auto mod_inv_5 = std::uint64_t(0xcccc'cccc'cccc'cccd);
      constexpr auto mod_inv_25 = mod_inv_5 * mod_inv_5;

      int s = 0;
      while (true) {
        auto q = bits::rotr(n * mod_inv_25, 2);
        if (q <= std::numeric_limits<std::uint64_t>::max() / 100) {
          n = q;
          s += 2;
        }
        else {
          break;
        }
      }
      auto q = bits::rotr(n * mod_inv_5, 1);
      if (q <= std::numeric_limits<std::uint64_t>::max() / 10) {
        n = q;
        s |= 1;
      }

      return s;
    }
  }

  static compute_mul_result compute_mul(
      carrier_uint u, cache_entry_type const &cache) noexcept {
    if constexpr (std::is_same_v<format, ieee754_binary32>) {
      auto r = wuint::umul96_upper64(u, cache);
      return {carrier_uint(r >> 32), carrier_uint(r) == 0};
    }
    else {
      static_assert(std::is_same_v<format, ieee754_binary64>);
      auto r = wuint::umul192_upper128(u, cache);
      return {r.high(), r.low() == 0};
    }
  }

  static constexpr std::uint32_t compute_delta(cache_entry_type const &cache,
                                               int beta) noexcept {
    if constexpr (std::is_same_v<format, ieee754_binary32>) {
      return std::uint32_t(cache >> (cache_bits - 1 - beta));
    }
    else {
      static_assert(std::is_same_v<format, ieee754_binary64>);
      return std::uint32_t(cache.high() >> (carrier_bits - 1 - beta));
    }
  }

  static compute_mul_parity_result compute_mul_parity(
      carrier_uint two_f, cache_entry_type const &cache, int beta) noexcept {
    assert(beta >= 1);
    assert(beta < 64);

    if constexpr (std::is_same_v<format, ieee754_binary32>) {
      auto r = wuint::umul96_lower64(two_f, cache);
      return {((r >> (64 - beta)) & 1) != 0,
              std::uint32_t(r >> (32 - beta)) == 0};
    }
    else {
      static_assert(std::is_same_v<format, ieee754_binary64>);
      auto r = wuint::umul192_lower128(two_f, cache);
      return {((r.high() >> (64 - beta)) & 1) != 0,
              ((r.high() << beta) | (r.low() >> (64 - beta))) == 0};
    }
  }

  static constexpr carrier_uint compute_left_endpoint_for_shorter_interval_case(
      cache_entry_type const &cache, int beta) noexcept {
    if constexpr (std::is_same_v<format, ieee754_binary32>) {
      return carrier_uint((cache - (cache >> (significand_bits + 2))) >>
                          (cache_bits - significand_bits - 1 - beta));
    }
    else {
      static_assert(std::is_same_v<format, ieee754_binary64>);
      return (cache.high() - (cache.high() >> (significand_bits + 2))) >>
             (carrier_bits - significand_bits - 1 - beta);
    }
  }

  static constexpr carrier_uint
  compute_right_endpoint_for_shorter_interval_case(
      cache_entry_type const &cache, int beta) noexcept {
    if constexpr (std::is_same_v<format, ieee754_binary32>) {
      return carrier_uint((cache + (cache >> (significand_bits + 1))) >>
                          (cache_bits - significand_bits - 1 - beta));
    }
    else {
      static_assert(std::is_same_v<format, ieee754_binary64>);
      return (cache.high() + (cache.high() >> (significand_bits + 1))) >>
             (carrier_bits - significand_bits - 1 - beta);
    }
  }

  static constexpr carrier_uint compute_round_up_for_shorter_interval_case(
      cache_entry_type const &cache, int beta) noexcept {
    if constexpr (std::is_same_v<format, ieee754_binary32>) {
      return (carrier_uint(cache >>
                           (cache_bits - significand_bits - 2 - beta)) +
              1) /
             2;
    }
    else {
      static_assert(std::is_same_v<format, ieee754_binary64>);
      return ((cache.high() >> (carrier_bits - significand_bits - 2 - beta)) +
              1) /
             2;
    }
  }

  static constexpr bool is_right_endpoint_integer_shorter_interval(
      int exponent) noexcept {
    return exponent >= case_shorter_interval_right_endpoint_lower_threshold &&
           exponent <= case_shorter_interval_right_endpoint_upper_threshold;
  }

  static constexpr bool is_left_endpoint_integer_shorter_interval(
      int exponent) noexcept {
    return exponent >= case_shorter_interval_left_endpoint_lower_threshold &&
           exponent <= case_shorter_interval_left_endpoint_upper_threshold;
  }
};

////////////////////////////////////////////////////////////////////////////////////////
// Policy holder.
////////////////////////////////////////////////////////////////////////////////////////

namespace policy_impl {
// The library will specify a list of accepted kinds of policies and their
// defaults, and the user will pass a list of policies. The aim of helper
// classes/functions here is to do the following:
//   1. Check if the policy parameters given by the user are all valid; that
//   means,
//      each of them should be of the kinds specified by the library.
//      If that's not the case, then the compilation fails.
//   2. Check if multiple policy parameters for the same kind is specified by
//   the user.
//      If that's the case, then the compilation fails.
//   3. Build a class deriving from all policies the user have given, and also
//   from
//      the default policies if the user did not specify one for some kinds.
// A policy belongs to a certain kind if it is deriving from a base class.

// For a given kind, find a policy belonging to that kind.
// Check if there are more than one such policies.
enum class policy_found_info { not_found, unique, repeated };
template <class Policy, policy_found_info info>
struct found_policy_pair {
  using policy = Policy;
  static constexpr auto found_info = info;
};

template <class Base, class DefaultPolicy>
struct base_default_pair {
  using base = Base;

  template <class FoundPolicyInfo>
  static constexpr FoundPolicyInfo get_policy_impl(FoundPolicyInfo) {
    return {};
  }
  template <class FoundPolicyInfo, class FirstPolicy,
            class... RemainingPolicies>
  static constexpr auto get_policy_impl(FoundPolicyInfo, FirstPolicy,
                                        RemainingPolicies... remainings) {
    if constexpr (std::is_base_of_v<Base, FirstPolicy>) {
      if constexpr (FoundPolicyInfo::found_info ==
                    policy_found_info::not_found) {
        return get_policy_impl(
            found_policy_pair<FirstPolicy, policy_found_info::unique>{},
            remainings...);
      }
      else {
        return get_policy_impl(
            found_policy_pair<FirstPolicy, policy_found_info::repeated>{},
            remainings...);
      }
    }
    else {
      return get_policy_impl(FoundPolicyInfo{}, remainings...);
    }
  }

  template <class... Policies>
  static constexpr auto get_policy(Policies... policies) {
    return get_policy_impl(
        found_policy_pair<DefaultPolicy, policy_found_info::not_found>{},
        policies...);
  }
};
template <class... BaseDefaultPairs>
struct base_default_pair_list {};

// Check if a given policy belongs to one of the kinds specified by the library.
template <class Policy>
constexpr bool check_policy_validity(Policy, base_default_pair_list<>) {
  return false;
}
template <class Policy, class FirstBaseDefaultPair,
          class... RemainingBaseDefaultPairs>
constexpr bool check_policy_validity(
    Policy, base_default_pair_list<FirstBaseDefaultPair,
                                   RemainingBaseDefaultPairs...>) {
  return std::is_base_of_v<typename FirstBaseDefaultPair::base, Policy> ||
         check_policy_validity(
             Policy{}, base_default_pair_list<RemainingBaseDefaultPairs...>{});
}

template <class BaseDefaultPairList>
constexpr bool check_policy_list_validity(BaseDefaultPairList) {
  return true;
}

template <class BaseDefaultPairList, class FirstPolicy,
          class... RemainingPolicies>
constexpr bool check_policy_list_validity(
    BaseDefaultPairList, FirstPolicy, RemainingPolicies... remaining_policies) {
  return check_policy_validity(FirstPolicy{}, BaseDefaultPairList{}) &&
         check_policy_list_validity(BaseDefaultPairList{},
                                    remaining_policies...);
}

// Build policy_holder.
template <bool repeated_, class... FoundPolicyPairs>
struct found_policy_pair_list {
  static constexpr bool repeated = repeated_;
};

template <class... Policies>
struct policy_holder : Policies... {};

template <bool repeated, class... FoundPolicyPairs, class... Policies>
constexpr auto make_policy_holder_impl(
    base_default_pair_list<>,
    found_policy_pair_list<repeated, FoundPolicyPairs...>, Policies...) {
  return found_policy_pair_list<repeated, FoundPolicyPairs...>{};
}

template <class FirstBaseDefaultPair, class... RemainingBaseDefaultPairs,
          bool repeated, class... FoundPolicyPairs, class... Policies>
constexpr auto make_policy_holder_impl(
    base_default_pair_list<FirstBaseDefaultPair, RemainingBaseDefaultPairs...>,
    found_policy_pair_list<repeated, FoundPolicyPairs...>,
    Policies... policies) {
  using new_found_policy_pair =
      decltype(FirstBaseDefaultPair::get_policy(policies...));

  return make_policy_holder_impl(
      base_default_pair_list<RemainingBaseDefaultPairs...>{},
      found_policy_pair_list < repeated ||
          new_found_policy_pair::found_info == policy_found_info::repeated,
      new_found_policy_pair, FoundPolicyPairs... > {}, policies...);
}

template <bool repeated, class... RawPolicies>
constexpr auto convert_to_policy_holder(found_policy_pair_list<repeated>,
                                        RawPolicies...) {
  return policy_holder<RawPolicies...>{};
}

template <bool repeated, class FirstFoundPolicyPair,
          class... RemainingFoundPolicyPairs, class... RawPolicies>
constexpr auto convert_to_policy_holder(
    found_policy_pair_list<repeated, FirstFoundPolicyPair,
                           RemainingFoundPolicyPairs...>,
    RawPolicies... policies) {
  return convert_to_policy_holder(
      found_policy_pair_list<repeated, RemainingFoundPolicyPairs...>{},
      typename FirstFoundPolicyPair::policy{}, policies...);
}

template <class BaseDefaultPairList, class... Policies>
constexpr auto make_policy_holder(BaseDefaultPairList, Policies... policies) {
  static_assert(
      check_policy_list_validity(BaseDefaultPairList{}, Policies{}...),
      "jkj::dragonbox: an invalid policy is specified");

  using policy_pair_list = decltype(make_policy_holder_impl(
      BaseDefaultPairList{}, found_policy_pair_list<false>{}, policies...));

  static_assert(!policy_pair_list::repeated,
                "jkj::dragonbox: each policy should be specified at most once");

  return convert_to_policy_holder(policy_pair_list{});
}
}  // namespace policy_impl
}  // namespace detail

////////////////////////////////////////////////////////////////////////////////////////
// The interface function.
////////////////////////////////////////////////////////////////////////////////////////

template <class Float, class FloatTraits = default_float_traits<Float>,
          class... Policies>
JKJ_FORCEINLINE JKJ_SAFEBUFFERS auto to_decimal(
    signed_significand_bits<Float, FloatTraits> signed_significand_bits,
    unsigned int exponent_bits, Policies... policies) noexcept {
  // Build policy holder type.
  using namespace detail::policy_impl;
  using policy_holder = decltype(make_policy_holder(
      base_default_pair_list<
          base_default_pair<sign::base, sign::return_sign>,
          base_default_pair<trailing_zero::base, trailing_zero::remove>,
          base_default_pair<decimal_to_binary_rounding::base,
                            decimal_to_binary_rounding::nearest_to_even>,
          base_default_pair<binary_to_decimal_rounding::base,
                            binary_to_decimal_rounding::to_even>,
          base_default_pair<cache::base, cache::full>>{},
      policies...));

  using return_type = decimal_fp<typename FloatTraits::carrier_uint,
                                 policy_holder::return_has_sign,
                                 policy_holder::report_trailing_zeros>;

  return_type ret = policy_holder::delegate(
      signed_significand_bits,
      [exponent_bits, signed_significand_bits](auto interval_type_provider) {
        using format = typename FloatTraits::format;
        constexpr auto tag = decltype(interval_type_provider)::tag;

        auto two_fc = signed_significand_bits.remove_sign_bit_and_shift();
        auto exponent = int(exponent_bits);

        if constexpr (tag == decimal_to_binary_rounding::tag_t::to_nearest) {
          // Is the input a normal number?
          if (exponent != 0) {
            exponent += format::exponent_bias - format::significand_bits;

            // Shorter interval case; proceed like Schubfach.
            // One might think this condition is wrong, since when exponent_bits
            // == 1 and two_fc == 0, the interval is actually regular. However,
            // it turns out that this seemingly wrong condition is actually
            // fine, because the end result is anyway the same.
            //
            // [binary32]
            // (fc-1/2) * 2^e = 1.175'494'28... * 10^-38
            // (fc-1/4) * 2^e = 1.175'494'31... * 10^-38
            //    fc    * 2^e = 1.175'494'35... * 10^-38
            // (fc+1/2) * 2^e = 1.175'494'42... * 10^-38
            //
            // Hence, shorter_interval_case will return 1.175'494'4 * 10^-38.
            // 1.175'494'3 * 10^-38 is also a correct shortest representation
            // that will be rejected if we assume shorter interval,
            // but 1.175'494'4 * 10^-38 is closer to the true value so it
            // doesn't matter.
            //
            // [binary64]
            // (fc-1/2) * 2^e = 2.225'073'858'507'201'13... * 10^-308
            // (fc-1/4) * 2^e = 2.225'073'858'507'201'25... * 10^-308
            //    fc    * 2^e = 2.225'073'858'507'201'38... * 10^-308
            // (fc+1/2) * 2^e = 2.225'073'858'507'201'63... * 10^-308
            //
            // Hence, shorter_interval_case will return 2.225'073'858'507'201'4
            // * 10^-308. This is indeed of the shortest length, and it is the
            // unique one closest to the true value among valid representations
            // of the same length.
            static_assert(std::is_same_v<format, ieee754_binary32> ||
                          std::is_same_v<format, ieee754_binary64>);

            if (two_fc == 0) {
              return decltype(interval_type_provider)::
                  invoke_shorter_interval_case(
                      signed_significand_bits,
                      [exponent](auto... additional_args) {
                        return detail::impl<Float, FloatTraits>::
                            template compute_nearest_shorter<
                                return_type,
                                typename decltype(interval_type_provider)::
                                    shorter_interval_type,
                                typename policy_holder::trailing_zero_policy,
                                typename policy_holder::
                                    binary_to_decimal_rounding_policy,
                                typename policy_holder::cache_policy>(
                                exponent, additional_args...);
                      });
            }

            two_fc |= (decltype(two_fc)(1) << (format::significand_bits + 1));
          }
          // Is the input a subnormal number?
          else {
            exponent = format::min_exponent - format::significand_bits;
          }

          return decltype(interval_type_provider)::invoke_normal_interval_case(
              signed_significand_bits,
              [two_fc, exponent](auto... additional_args) {
                return detail::impl<Float, FloatTraits>::
                    template compute_nearest_normal<
                        return_type,
                        typename decltype(interval_type_provider)::
                            normal_interval_type,
                        typename policy_holder::trailing_zero_policy,
                        typename policy_holder::
                            binary_to_decimal_rounding_policy,
                        typename policy_holder::cache_policy>(
                        two_fc, exponent, additional_args...);
              });
        }
        else if constexpr (tag == decimal_to_binary_rounding::tag_t::
                                      left_closed_directed) {
          // Is the input a normal number?
          if (exponent != 0) {
            exponent += format::exponent_bias - format::significand_bits;
            two_fc |= (decltype(two_fc)(1) << (format::significand_bits + 1));
          }
          // Is the input a subnormal number?
          else {
            exponent = format::min_exponent - format::significand_bits;
          }

          return detail::impl<Float>::template compute_left_closed_directed<
              return_type, typename policy_holder::trailing_zero_policy,
              typename policy_holder::cache_policy>(two_fc, exponent);
        }
        else {
          static_assert(
              tag == decimal_to_binary_rounding::tag_t::right_closed_directed);

          bool shorter_interval = false;

          // Is the input a normal number?
          if (exponent != 0) {
            if (two_fc == 0 && exponent != 1) {
              shorter_interval = true;
            }
            exponent += format::exponent_bias - format::significand_bits;
            two_fc |= (decltype(two_fc)(1) << (format::significand_bits + 1));
          }
          // Is the input a subnormal number?
          else {
            exponent = format::min_exponent - format::significand_bits;
          }

          return detail::impl<Float>::template compute_right_closed_directed<
              return_type, typename policy_holder::trailing_zero_policy,
              typename policy_holder::cache_policy>(two_fc, exponent,
                                                    shorter_interval);
        }
      });

  policy_holder::handle_sign(signed_significand_bits, ret);
  return ret;
}

template <class Float, class FloatTraits = default_float_traits<Float>,
          class... Policies>
JKJ_FORCEINLINE JKJ_SAFEBUFFERS auto to_decimal(Float x,
                                                Policies... policies) noexcept {
  auto const br = float_bits<Float, FloatTraits>(x);
  auto const exponent_bits = br.extract_exponent_bits();
  auto const s = br.remove_exponent_bits(exponent_bits);
  assert(br.is_finite());

  return to_decimal<Float, FloatTraits>(s, exponent_bits, policies...);
}
}  // namespace jkj::dragonbox

#undef JKJ_FORCEINLINE
#undef JKJ_SAFEBUFFERS
#undef JKJ_DRAGONBOX_HAS_BUILTIN

#endif
